Moduli of Galois Representations

A dissertation presented

by

Carl William Wang Erickson

to

The Department of Mathematics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Mathematics

Harvard University

Cambridge, Massachusetts

April 2013

© 2013 – Carl William Wang Erickson

All rights reserved.

Dissertation Advisor: Professor Mark Kisin Carl William Wang Erickson

Moduli of Galois Representations1

Abstract

The theme of this thesis is the study of moduli stacks of representations of an associativealgebra, with an eye toward continuous representations of profinite groups such as Galoisgroups. The central object of study is the geometry of the map ψ from the moduli stack ofrepresentations to the moduli scheme of pseudorepresentations. The first chapter culminatesin showing that ψ is very close to an adequate moduli space of Alper [Alp10]. In particular,ψ is universally closed. The second chapter refines the results of the first chapter. Inparticular, certain projective subschemes of the fibers of ψ are identified, generalizing asuggestion of Kisin [Kis09a, Remark 3.2.7]. The third chapter applies the results of the firsttwo chapters to moduli groupoids of continuous representations and pseudorepresentationsof profinite algebras. In this context, the moduli formal scheme of pseudorepresentations issemi-local, with each component Spf BD being the moduli of deformations of a given finitefield-valued pseudorepresentation D. Under a finiteness condition, it is shown that ψ is notonly formally finite type over Spf BD, but arises as the completion of a finite type algebraicstack over SpecBD. Finally, the fourth chapter extends Kisin’s construction [Kis08] of lociof coefficient spaces for p-adic local Galois representations cut out by conditions from p-adicHodge theory. The result is extended from the case that the coefficient ring is a completeNoetherian local ring to the more general case that the coefficient space is a Noetherianformal scheme.

1Version 1.1 of May 17, 2013.

iii

Contents

Acknowledgements vi

Chapter 1. Pseudorepresentations and Representations 11.1. Pseudorepresentations 21.1.1. Introduction to Pseudorepresentations 21.1.2. Polynomial Laws 51.1.3. Representability 81.1.4. Symmetric Tensor Algebras 161.1.5. Faithful Polynomial Laws 171.1.6. Multiplicative Polynomial Laws 181.1.7. Definition of Pseudorepresentations and Representability of the

Pseudorepresentation Functor 211.1.8. The Characteristic Polynomial 261.1.9. Universal Polynomial Identities 281.1.10. Work of Vaccarino, Donkin, Zubkov, and Procesi 321.1.11. A Direct Sum Operation on Pseudorepresentations 381.1.12. Relation to Pseudocharacters 411.2. Cayley-Hamilton Pseudorepresentations 421.2.1. Properties of Cayley-Hamilton Algebras 421.2.2. Background in PI Ring Theory 441.2.3. The Jacobson Radical of a Cayley-Hamilton Algebra 481.2.4. The Universal Cayley-Hamilton Algebra 501.3. Pseudorepresentations over Fields 511.3.1. Main Theorem 521.3.2. Semisimple k-algebras 541.3.3. Finite-Dimensional Cayley-Hamilton Algebras 581.3.4. Composition Factors of Field-Valued Pseudorepresentations 581.4. Moduli Spaces of Representations 611.4.1. Moduli Schemes and Algebraic Stacks 611.4.2. Mapping Algebraic Stacks of Representations to the Moduli Scheme of

Pseudorepresentations 641.4.3. Representations Factor through the Universal Cayley-Hamilton Algebra 651.4.4. Representations of Groups into Affine Group Schemes 671.5. Geometric Invariant Theory of Representations 701.5.1. Alper’s Theory of Adequate Moduli Spaces 701.5.2. Geometric Invariant Theory on Rep 731.5.3. Work of Kraft, Richardson, et al. on Orbits of the Adjoint Action on

Representations 731.5.4. The GIT quotient and PsRd

R are Almost Isomorphic 75

Chapter 2. Local Study of Pseudorepresentations 802.1. Pseudorepresentations over Local Rings 812.1.1. Deformation Theory Setup 812.1.2. Tangent Spaces of the Pseudorepresentation Functor 822.1.3. Cayley-Hamilton Pseudorepresentations over Local Rings 84

iv

2.2. Fibers of ψ 862.2.1. King’s Result on Quiver Representation Moduli 882.2.2. Finite Dimensional Algebras and Path Algebras 942.2.3. Examples of Projective Moduli Spaces 982.2.4. Deformation of Ample Line Bundles 1012.3. Multiplicity Free Pseudorepresentations 1032.3.1. Generalized Matrix Algebras 1042.3.2. Trace Representations and Adapted Representations 1052.3.3. Invariant Theory of Adapted Representations 106

Chapter 3. Representations and Pseudorepresentations of Profinite Algebras 1113.1. Pseudorepresentations of Profinite Algebras 1113.1.1. Pro-discrete Topological Notions 1123.1.2. Continuous Pseudorepresentations of Profinite Algebras 1143.1.3. The Φp Finiteness Condition on Profinite Groups 1163.1.4. Continuous Deformations of a Finite Field-Valued Pseudorepresentation 1173.1.5. Finiteness Condition ΦD 1203.1.6. Pseudorepresentations valued in Formal Schemes 1223.2. Moduli of Representations of a Profinite Algebra 1263.2.1. Groupoids of Representations 1273.2.2. Finiteness Results 1283.2.3. Universality Results 1313.2.4. Representability Results 1333.2.5. Consequences of Algebraization 134

Chapter 4. p-adic Hodge Theory in Group-Theoretic Families 1364.1. Changes in Notation from [Kis08] 1374.2. Background for Representations of Bounded E-height (§§4.3-4.5) 137

4.3. Families of Etale ϕ-modules 1384.4. Functors of Lattices and Affine Grassmanians 1424.5. Universality of M in Characteristic 0 1504.6. Background for Families of Filtered (ϕ,N)-modules (§§4.7-4.12) 1534.7. Families of (ϕ,N)-modules over the Open Unit Disk 1574.8. Period Rings in Families 1604.9. Period Maps 1644.10. Moduli Space of Semistable Representations 1704.11. Hodge Type 1724.12. Galois Type 1734.13. Final Remarks 179

Appendix A. A Remark on Projective Morphisms 181

Appendix. Bibliography 182

v

Acknowledgements

I am thankful for many people who have made the past five years such an engaging andstimulating experience. Firstly, I warmly thank my advisor, Mark Kisin, for his remarkablededication and patience in advising me and introducing me to many new mathematical ideas,including the suggestion to pursue the line of inquiry in this dissertation. His mentorshipand guidance has been invaluable. I would also like to thank Dick Gross, Barry Mazur,and Joe Harris for their advice and encouragement. The mathematical debt to both GaetanChenevier and Mark Kisin will be clear to the reader. I also thank Brian Conrad andBrandon Levin for useful conversations toward the completion of this work, and gratefullyrecognize the support of the National Science Foundation in the form of a Graduate ResearchFellowship.

I would like to thank many colleagues and friends that I shared my graduate schoolyears with: Nathan Kaplan, David Hansen, Brandon Levin, Jack Thorne, Nathan Pflueger,Hansheng Diao, and many others added both mathematical stimulation and personal warmthto my graduate school experience. I am also very thankful for friends at Highrock Churchand the Harvard Graduate Christian Community, who have supported me and enriched mylife throughout my time in graduate school.

There are many teachers and mentors who encouraged me in my pursuit of mathematics,and I particularly want to recognize and thank my high school mentor Dale Snider. It isalso a pleasure to thank my undergraduate advisors Ben Brubaker and Ravi Vakil for theirconscientious and enthusiastic mentorship.

Many thanks go to my parents for supporting me as a person and as a mathematician,and to Scott for his companionship.

My dearest thanks go to Anna: thank you for your love and friendship.

vi

CHAPTER 1

Pseudorepresentations and Representations

In this chapter, we develop the notion of a pseudorepresentation following Chenevier[Che11], who calls them determinants. Essentially, a pseudorepresentation in the data of thecharacteristic polynomials of a representation. We make it a goal in §1.1 is to give a thoroughexposition of the theory of pseudorepresentations. We emphasize that much of this contentis due to Chenever; he, in turn, synthesizes and builds on work of Roby, Vaccarino, Donkin,Zubkov, Procesi, and others. We will make these attributions clear. In particular, §§1.1-1.3are due in large part to Chenevier, and our presentation follows his, adding detail to thesources he draws on. Our original contributions in these sections consist of the applicationof polynomial identity ring theory, which we begin to discuss in §1.2.2.

Starting in §1.4, we define and study moduli stacks of representations. A representationinduces a pseudorepresentation, so that there is a natural morphism from moduli stacks ofrepresentations to the moduli scheme of pseudorepresentation. Our task is to study the ge-ometry of this morphism, which we call ψ. Our main result in this chapter, Theorem 1.5.4.2,is that ψ is very nearly an adequate moduli space. Adequate moduli spaces, a notion dueto Alper [Alp10, Alp08], are introduced in §1.5.1. They are meant to generalize a situationcommonly arising in geometric invariant theory (GIT); they are basically “isomorphismsminus representability,” having important properties of both proper morphisms and affinemorphisms (cf. Remark 1.5.1.5). Indeed, a more precise way of stating our result is that themoduli space of pseudorepresentations differs by at most a finite universal homeomorphismfrom the GIT quotient of the moduli scheme of framed representations by the natural actionof conjugation.

The controlling idea is that the moduli scheme of pseudorepresentations is a concretereplacement for the GIT quotient of the moduli scheme of representations by the actionof conjugation. As pseudorepresentations have a sensible functorial definition, the moduliproblem of pseudorepresentations is representable by an affine scheme. This is what we meanby “concrete.” On the other hand, the GIT quotient of a moduli scheme by a reductive groupof natural automorphisms has a priori a moduli-theoretic interpretation only for its functorof geometric points, even though it is a scheme (cf. Theorem 1.5.1.4(2), Remarks 1.5.1.6 and1.5.2.3). We lack a functor of points because the universal property of a quotient addressesmoprhisms out of the quotient instead of morphisms to the quotient. The moduli space ofpseudorepresentations is useful because it nearly attains both universal properties.

Later, in Chapter 2, we will improve Theorem 1.5.4.2, identifying certain loci in the baseover which ψ is an adequate moduli space. However, we expect that it is always an adequatemoduli space; Corollary 2.3.3.9 will provide some evidence.

In this chapter, we also begin to see what the concreteness of the moduli problem of pseu-dorepresentations affords to us. The main thing we achieve in this chapter is the demon-stration of some finiteness properties of representations that are visible when one studiesmoduli spaces of representations relative to the moduli space of pseudorepresentations. We

1

accomplish this by applying the notion of a Cayley-Hamilton pseudorepresentation and us-ing results from polynomial identity ring theory to study the category of Cayley-Hamiltonrepresentations (see §1.2). In particular, if a non-commutative algebra is finitely generatedover a commutative Noetherian ring, we show in Theorem 1.4.3.1 that all of its d-dimensionalrepresentations canonically factor through a single algebra that is finite as a module overits center. This line of thought will be followed in Chapter 3, when we approach our maingoal of studying moduli spaces of representations and pseudorepresentations of profinitegroups and algebras in the category of formal schemes. In particular, we show that under asuitable finiteness condition on a profinite algebra, its moduli stacks of representations arealgebraizable over the moduli space of pseudorepresentations (cf. Corollary 3.2.4.3).

Notation and Conventions. Throughout this chapter, we will consider representationsand pseudorepresentations of non-commutative algebras over commutative rings. All ringsand algebras are associative, and they are unital except in some discussion of nil-algebrasin §1.2.2. Generally, we will use A for a commutative base ring of coefficients, R as a A-algebra whose representations we study, and B for a commutative A-algebra of coefficients.When a fixed base is needed to study functors and groupoids of representations and pseu-dorepresentations, we will use a commutative ring A, so that B are commutative A-algebras.Sometimes, we will also assume that A is Noetherian and that R is finitely generated inorder that the schemes paramterizing the representation and pseudorepresentation functorswill be finite type over SpecA. We may also use S as a base coefficient scheme. We willuse Γ for a group, often finitely generated, when we want to study group representations.Except for some study of the moduli of group scheme valued representations in §1.4.4, westudy the representations of Γ by studying the representations and pseudorepresentations ofR = A[Γ].

1.1. Pseudorepresentations

In this section, we give an introduction to pseudorepresentations, our goal being toprovide a thorough exposition of background material on pseudorepresentations. All ofthis material is due to Roby [Rob63, Rob80]1 and Chenevier [Che11, §§1-2]. Chenevieremphasizes that the main theorems (Theorem 1.3.1.1 and Theorem 2.1.3.3) “should not beconsidered as original, as they could probably be deduced from earlier works of Procesi viathe relation between determinants and generic matrices established by Vaccarino, Donkin,and Zubkov” [Che11, p. 4]. We will give these attributions as they appear. We note that wecall “pseudorepresentation” what Chenevier calls a “determinant.”

1.1.1. Introduction to Pseudorepresentations. Let A be a commutative ring andR be an A-algebra. We will give a preliminary (but accurate) definition of a pseudorepre-sentation that we will provide more theoretical context for in the sequel.

Definition 1.1.1.1. A d-dimensional pseudorepresentationD of R over A is the structureof an A-algebra on R and an association to each commutative A-algebra B to the map

DB : R⊗A B −→ B

satisfying the following conditions:

(1) DB is multiplicative and unit-preserving (but not necessarily additive),

1A helpful summary of Roby’s work appears in [BO78, Appendix A].

2

(2) DB is homogenous of degree d, i.e.

∀ b ∈ B, ∀ x ∈ R⊗A B,DB(bx) = bdDB(x),

(3) D is functorial on A-algebras, i.e. for any map B → B′ of commutative A-algebras,the diagram

R⊗A BDB //

B

R⊗A B′

DB′ // B′

commutes.

We write D : R→ A for such data, although this data is much more than a map from R toA.

Notation. If R is an A-algebra and B is a commutative A-algebra, we denote byPsRd

R(B) the set of d-dimensional pseudorepresentations of R ⊗A B over B. We will soonsee that there is a natural structure of a functor on this map from A-algebras to sets.

The main interest in pseudorepresentations comes from their relation to representationsthrough the characteristic polynomial of a representation. Indeed, a d-dimensional represen-tation ρ : R → Md(A) of R over A induces a d-dimensional pseudorepresentation D of Rover A as follows. For any commutative A-algebra B, let DB be the composition

(1.1.1.2) DB : R⊗A Bρ⊗AB−→ Md(A)⊗A B

∼−→Md(B)det−→ B

of the representation itself with the determinant map. We observe that

(1) DB is multiplicative, because ρ and det are multiplicative,(2) DB is homogenous of degree d because ρ is linear (homogenous of degree 1) and det

is homogenous of degree d, and(3) one can check that the functionality condition (3) holds.

This map from representations to pseudorepresentations is a bit abstract until one con-siders the characteristic polynomial associated to a pseudorepresentation. Of course, thischaracteristic polynomial exists even if the pseudorepresentation does not come from a rep-resentation.

Definition 1.1.1.3. Given a d-dimensional pseudorepresentation D : R −→ A of R overA, its characteristic polynomial function is

χ(·, t) : R −→ A[t]

r 7→ DA[t](t− r) = td − Λ1(r)td−1 + · · ·+ (−1)dΛd(r)

where Λi are maps R → A. We will sometimes write these maps as ΛDi when specificity is

required.

When we have a representation ρ : R → Md(A) and the induced pseudorepresentationD = D(ρ), a look at the definitions allows us to see that the characteristic polynomial ofr ∈ R under the representation ρ, that is, the polynomial

det(t · Id×d − ρ(r)) ∈ A[t],

is identical to the characteristic polynomial of the pseudorepresentation D. Therefore, apseudorepresentation retains at least as much information as the characteristic polynomialof a representation. Later, we will see that it has exactly this much information, i.e. a

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pseudorepresentation is characterized by its characteristic polynomial coefficients Λi (Corol-lary 1.1.9.15). It is useful to think of a pseudorepresentation as the data of characteristicpolynomial coefficient functions Λi on R satisfying the relations imposed by being the char-acteristic polynomial of a representation. However, we never have need to these relationsexplicit.

Remark 1.1.1.4. The construction of (1.1.1.2) works just as well when Md(A) is replacedby R, an Azumaya A-algebra of rank d2 over its center A, and det is replaced by the reducednorm R → A. This includes the case of the Azumaya algebra EndA(V ) for a projectiverank d A-module V . These three notions of representation – a homomorphism into a d-by-dmatrix algebra, a linear action on a rank d projective module, and a homomorphism intoan Azumaya algebra – are defined in Definition 1.4.1.1 and explored in §1.4. Note that eachnotion of representation listed includes the previous notions in the list. Profinite topologicalanalogues of these representations are given in Definition 3.2.1.1 and studied in Chapter 2.

It is well known that a semisimple representation of an algebra over an algebraicallyclosed field is characterized up to isomorphism by its characteristic polynomial, and there-fore also by its associated pseudorepresentation. This leads us to ask when we can reversethe map from representations to pseudorepresentations, and make a representation from apseudorepresentation. In fact, we describe in Theorem 1.3.1.1 a result of Chenevier: givenan algebraically closed A-field k, the induced map

semisimple d-dimensional representations of R⊗A k/ ∼↓

PsRdR(k)

is a bijection! This is an important fact, suggesting that “pseudorepresentations” deservetheir name: over an algebraically closed field, they are realizable as the determinant of arepresentation.

Let us overview the content of this section. Pseudorepresentations are, in fact, partic-ular cases of multiplicative polynomial laws, which is a notion due to Roby. Roby’s work[Rob63] on polynomial laws is reviewed for several of the next paragraphs, and we discusshis work on multiplicative polynomial laws starting in §1.1.6. Then, we follow Chenevier:pseudorepresentations are defined in §1.1.7, and a universal pseudorepresentation and theresulting moduli scheme are identified in Theorem 1.1.7.4. Then, in §§1.1.8-1.1.9, we exploreproperties of pseudorepresentations through their characteristic polynomials. In particular,“Amitsur’s formula” shows that a pseudorepresentation is characterized by its characteris-tic polynomial, and a certain “Cayley-Hamilton identity” holds for pseudorepresentations.Work of Vaccarino [Vac08] critical for these identities is reviewed in §1.1.10, along with con-tributions of Donkin and others that he build upon. The only original part of this section is§1.1.11, where we give a direct sum operation on pseudorepresentations that decategorifiesthe direct sum operation on representations. This makes the moduli space of pseudorepre-sentations of all dimensions a “scheme in commutative monoids.” We conclude this sectionin §1.1.12 with a discussion of an alternate notion of pseudorepresentation which we call“pseudocharacters” in order to indicate that they retain only the data of a trace function,while a pseudorepresentation consists of the entire characteristic polynomial. This notion ofpseudorepresentation is due to Taylor [Tay91] following a definition of Wiles [Wil88].

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1.1.2. Polynomial Laws. A d-dimensional determinant of R over A is a particularcase of a polynomial law that is homogenous of degree d and multiplicative. We will definethese terms and make a preliminary study of them. This content is originally due to Roby[Rob63, Rob80], and we are indebted to the exposition of Chenevier [Che11, §1]. The mainresult, Theorem 1.1.7.4, is that the functor of d-dimensional pseudorepresentations PsRd

R onA-algebras is representable. In particular, there exists a universal pseudorepresentation.

First we define a polynomial law.

Definition 1.1.2.1 ([Rob63, §1.2]). Let A be a commutative ring and let M,N be A-modules. A polynomial law P : M → N is the association to each commutative A-algebraB a set-theoretic map

PB : M ⊗A B → N ×A Bsuch that for any A-algebras B → B′, the diagram

(1.1.2.2) M ⊗A BPB //

N ⊗A B

M ⊗A B′

PB′ // N ⊗A B′

commutes. The set of polynomial laws from M to N is denoted PA(M,N).

Remark 1.1.2.3. A more sophisticated way of saying this is that if M is the quasi-coherent sheaf on the big Zariski site Sch/ SpecA associated to M , then a polynomial lawP : M → N is just a morphism (M)Set → (N)Set of the underlying sheaves of sets.

Example 1.1.2.4. Let M −→ N be a homomorphism of A-modules. Applying ⊗ABto this homomorphism for each A-algebra B defines a polynomial law. This is a linearpolynomial law. The set of linear polynomial laws in PA(M,N) is the image of HomA(M,N)under the mapping we have just described.

It is possible to apply to polynomial laws many of the notions applicable to conventionalpolynomial functions. For example, given A-modules M,N,P , we may compose pairs inPA(M,N) × PA(N,P ). Also, PA(M,N) is naturally an A-module: if P, P ′ ∈ PA(M,N),then setting (P + P ′)B to be PB + P ′B for each A-algebra B defines a valid polynomial law.Likewise, for a ∈ A, P ∈ PA(M,N), composing PB with the action of a on N ⊗A B for eachA-algebra B gives an A-module structure on PA(M,N).

Polynomials are sums of their homogenous components of each degree. In the same way,we can assign a degree to a polynomial law and define homogenous polynomial laws.

Definition 1.1.2.5. Let P ∈ PA(M,N) be a polynomial law between A-modules M,Nand let d ≥ 0. We call P homogenous of degree d if for all A-algebras B, all b ∈ B, and allx ∈M ⊗A B,

P (bx) = bdP (x).

We write PdA(M,N) for the set of polynomial laws of degree d.

We observe that PdA(M,N) is a sub-A-module of PA(M,N): if P, P ′ ∈ PdA(M,N), thentheir sum is homogenous of degree d, as is a scalar multiple. In fact, just as for conventionalpolynomials, a polynomial law can be decomposed into its homogenous components of eachdegree. Following [BO78, §A4], take P ∈ PA(M,N) and consider for any A-algebra B themap PB[t] : M⊗AB[t]→ N⊗AB[t]. For any x ∈M⊗AB, define P d

B(x) ∈ N⊗AB according

5

to the formulaPB[t](x⊗ t) =

∑d≥0

P d(x)td.

Since this is an element of N ⊗A B[t], P d(x) = 0 for sufficiently large d, i.e. this is a“polynomial” in t. One can then check that the P d

B are functorial in B, so that they definea P d ∈ PdA(M,N). Then by using the functionality of P under the map B[t]→ B, t 7→ 1 foreach A-algebra B, we see that PB(x) =

∑d≥0 P

dB(x) for all A-algebras B and all x ∈M⊗AB.

This decomposition into homogenous components is locally finite, in the sense that for anygiven element of M ⊗A B, the result is a polynomial. However, as B and x vary, thepolynomial “degree” may grow.

Remark 1.1.2.6. As noted in [BO78, Appendix A], there is a sort of analogy betweenhomogenous polynomial laws (of a given degree) and modular functions (of a given weight).

Example 1.1.2.7 ([Rob63, Proposition I.5]). When P ∈ PA(M,N) is homogenous ofdegree 0, this is a “constant” polynomial law, and amounts to n ∈ N such that PB(x) =n⊗ 1 ∈ N ⊗A B for all x ∈M ⊗A B.

Example 1.1.2.8 ([Rob63, §I.4]). Any linear polynomial law is homogenous of degree 1.Using the Yoneda Lemma, one can see that the map HomA(M,N)→ PA(M,N) defines an

isomorphism of A-modules HomA(M,N)∼→ P1

A(M,N).

Example 1.1.2.9 (cf. [Rob63, §II.3]). Let P ∈ P2A(M,N) be a homogenous degree 2

polynomial law. This gives rise to a bilinear form BP : M ⊕M → N by setting

BP (m,m′) = PA(m+m′)− PA(m)− PA(m′).

One can check that this is bilinear by observing that BP is the t1t2-coefficient of

PA[t1,t2](m1 ⊗ t1 +m2 ⊗ t2);

while this is just the standard association of a bilinear form to a quadratic form, see Definition1.1.2.14 to set this in the context of polynomial laws. In fact, P is characterized by PA, andis characterized by BP if 2 is not a zero divisor in A. Conversely, if Q : M → N is a quadraticmap, i.e. Q(am) = a2Q(m) for all a ∈ A,m ∈M , and if BQ constructed as above is bilinear,

then there exists a unique polynomial law Q ∈ P2A(M,N) such that QA = Q. This is proved

in [Rob63, Proposition II.1].

The example above shows that a polynomial law of degree two P ∈ P 2(M,N) is actuallydetermined by PA, i.e. PA determines PB for all A-algebras B, cf. [Rob63, Proposition II.1].However, this is not necessarily the case in general, as the following example shows.

Example 1.1.2.10 ([Che11, Example 1.2(iii)]). The Frobenius automorphism of the fieldA = F2 of 2 elements can be used to find a polynomial law of degree 3 P ∈ P3

A(M,N) notdetermined by PA. Let M be a two-dimensional A-vector space and let N = A. Choosea basis X, Y of HomA(M,A). Then the A-polynomial law P : M → N defined byXY 2 −X2Y is homogenous of degree q + 1. Clearly P 6= 0, but PA is the zero map.

On the other hand, there are general conditions for a polynomial law to be determinedby its restriction PA : M → N to M .

Proposition 1.1.2.11 ([Rob63, Proposition I.8]). If the ring A is an infinite cardinalitydomain and if for every 0 6= x ∈ N there exists some A-linear map β : N → A such thatβ(x) 6= 0, then any P ∈ PA(M,N) is determined by PA.

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For example, any free A-module N satisfies the condition on N in the statement.One can isolate a single “coefficient” of a polynomial through the following procedure.

Definition 1.1.2.12. Let P ∈ PA(M,N) and choose integers p ≥ 1 and let α =(α1, . . . , αp) be a p-tuple of non-negative integers. Then if A[t1, . . . , tp] is the free polynomialalgebra over A in p variables t1, . . . , tp, write P [α] : M⊕p to N as the following function. For(m1, . . . ,mp) ∈Mp, P [α](m1, . . . ,mp) is the coefficient of tα1

1 tα22 · · · t

αpp in

PA[t1,...,tp]

(p∑i=1

miti

).

Note that by applying the homogeneity condition, we may quickly see that if P ∈PdA(M,N), then P [α] 6≡ 0 =⇒

∑pi=1 αi = d.

Remark 1.1.2.13 (cf. [Rob63, §II.2]). If P ∈ PdA(M,N) and p = d, i.e. if α is the d-tuple(1, . . . , 1), then P [α] is a multilinear function M⊗d → N . Roby calls this the “completepolarization” of a homogenous polynomial law. For d = 2, we have already seen this inExample 1.1.2.9 above. We readily observe that this multilinear function is symmetric,i.e. commutes with the action of Sd. When d! is invertible in A, this defines a bijectionbetween homogenous polynomial laws of degree d and symmetric multilinear functions fromM⊗d to N . This means that when PdA(M,−) is representable by SymdM , which we note

is a quotient of M⊗d. The universal object (corresponding to SymdMid−→ SymdM) is the

d-dimensional homogenous polynomial law

m 7→ m⊗ · · · ⊗m/p! ∈ PdA(M, SymdM).

However, since the main goal of introducing pseudorepresentations in place of pseudochar-acters2 is to allow the characteristic of coefficient rings of representations to be arbitrary, wewant a theory without this weakness. In fact, there is a universal object for PdA(M,−) forarbitrary A and M , as we will see in the sequel.

We are interested in a simple, explicit set of functions that characterize a polynomiallaw. These are the “coefficients” of the polynomial law.

Definition 1.1.2.14. Let P ∈ PA(M,N). Then for any choice of positive integer n ≥ 1,any choice of m1, . . . ,mn ∈ M , and any ordered n-tuple of integers α = (α1, . . . , αn), setP [α] : Mn −→ N by

PA[t1,...,tn](n∑i=1

mi ⊗ ti) =∑α

P [α](m1, . . . ,mn)tα,

where tα :=∏n

i=1 tαi .

Definition 1.1.2.15. Let Idn denote the set of tuples α = (α1, . . . , αn) of non-negativeintegers such that their sum is d. We will use the notation Id when the size n of the tupleis clear from the context.

Often, we will use Id will refer to d-tuples, since at most d entries of an element of Idn arenon-zero and the data, such as P [α], labeled by a choice of an element of Idn are completelydetermined by the data labeled by elements of Idd .

2In short, pseudocharacters keep track of the trace function of a representation, while pseudorepresentationskeep track of the entire characteristic polynomial. See §1.1.12 for a discussion of pseudocharacters and thehistory of notions of pseudorepresentations.

7

The following proposition shows that the coefficients of a polynomial law share analogousproperties to the coefficients of a polynomial.

Proposition 1.1.2.16 (cf. [Rob63, Theorem I.1]). With P , α ∈ Idn, and P [α] as in thedefinition above, then

(1) For any m1, . . . ,mn ∈ Mn, there are only finitely many n-tuples of non-negativeintegers α such that P [α](m1, . . . ,mn) 6= 0.

(2) If m1, . . . ,mn generates M , then P [α](m1, . . . ,mn) characterizes P , as α varies overall n-tuples of non-negative integers.

(3) If P is homogenous of degree d ≥ 0, then P is characterized by the degree d homoge-nous functions P [α] | α ∈ Idd, and P [α] ≡ 0 if

∑α αi 6= d.

(4) If P is homogenous of degree d, then P is characterized by the degree d homogenousfunction PA[t1,...,td] : R⊗A A[t1, . . . , td]→ A[t1, . . . , td].

Proof. Parts (1) and (2) are proved in [Rob63, Theorem I.1]. Part (4) clearly followsfrom (2) and (3).

Let us prove (3). By part (2), it will suffice to show that if α ∈ Id′ where d 6= d′

and P ∈ PdA(M,N), then P [α] = 0. Say that we have a counterexample α ∈ Ind′ , so thatthere exists some n-tuple (x1, . . . , xn) ∈ Mn such that P [α](x1, . . . , xn) 6= 0. Then if t is anindeterminant,

(1.1.2.17)

td · P (n∑i=1

miti) = P (t ·n∑i=1

miti)

= P (n∑i=1

mi(tti)

=∑D≥0

∑α∈InD

P [α](m1, . . . ,mn) · tD · tα

where we recall that tα stands for∏n

i=1 tαii . Setting D = d′ and (m1, . . . ,mn) = (x1, . . . , xn)

and α = α, and expanding the expression above in terms of monomials in the variablest, t1, . . . , tn, we have that there is a nonzero term in the final line of (1.1.2.17) where thedegree of t is not d. This violates the homogeneity expressed in the first line of (1.1.2.17).

1.1.3. Representability. Now we assemble notions needed to specify the A-modulerepresenting the functors N 7→ PA(M,N) and N 7→ PdA(M,N) for d ≥ 0. Firstly we definethe divided power A-algebra ΓA(M) of M along with its graded component A-modulesΓdA(M), and construct a degree d homogenous polynomial law

Ld : M −→ ΓdA(M).

This will turn out to be the representing A-module and universal object for PdA(M,−).

Definition 1.1.3.1. Let M be an A-module. The commutative A-algebra ΓA(M) is thequotient algebra of the polynomial algebra generated by the symbols m[i] where m ∈M andi ≥ 0, subject to the relations

(1) m[0] = 1 for m ∈M ,(2) (am)[i] = aim[i] for a ∈ A,m ∈M, i ≥ 0,(3) m[i]m[j] = i!j!

(i+j)!m[i+j] for m ∈M, i, j ≥ 0, and

8

(4) (m+m′)[i] =∑

p+q=im[p]m′[q].

Assigning to m[i] the degree i, we see that as the relations form a homogenous ideal, ΓA(M)is a graded A-algebra under this grading, and we denote by ΓdA(M) the dth graded piece, so

ΓA(M) ∼=⊕d≥0

ΓdA(M).

Definition 1.1.3.2. We define the universal degree d homogenous polynomial law Ld ∈P d(M,ΓdA(M)) by the maps

LdB : M ⊗A B −→ ΓA(M)⊗A B ∼= ΓB(M ⊗A B)

m⊗ b 7→ m[i] ⊗ bi ∼= (bm)[i].

The relations above show that this map is well-defined, functorial in B, and degree d ho-mogenous.

We think of the element m[i] as “mi/i!,” even though i! may not be invertible in A.

Remark 1.1.3.3. This graded algebra is called the divided power algebra of M becausethe ideal Γ+

A(M) ⊂ ΓA(M) of positive degree elements is a divided power ideal for ΓA(M)with divided power structure γ = (γi) characterized by the property γi(m

[1]) = m[i] for allm ∈ M, i ≥ 0 [BO78, Theorem A9]. This algebra has a special universal property amongA-algebras with a divided power ideal [BO78, Theorem 3.9].

The universality of Ld among degree d homogenous polynomial laws out of M is sum-marized by this theorem.

Theorem 1.1.3.4 ([Rob63, Theorem IV.1]). Let M,N be two A-modules and let d ≥ 0.There is a canonical isomorphism

(1.1.3.5) HomA(ΓdA(M), N)∼−→ PdA(M,N)

given by sending f ∈ HomA(ΓdA(M), N) to its composition f Ld with the universal degree dhomogenous polynomial law Ld : M → ΓdA(M).

We can already see that the map (1.1.3.5) exists, by composing Ld with a linear mapΓdA(M)→ N . In order to show it is a bijection as the theorem claims, we need to introducetwo notions: a natural functor that the A-algebra ΓA(M) represents, and the notion of thederivative of a polynomial law. In this we follow [BO78, Appendix A].

In order to prove Theorem 1.1.3.4, we introduce two tools. The first is the exp functor.

Definition 1.1.3.6. Let B be a commutative A-algebra. Let exp(B) be the followingB-module, a subgroup of the abelian group of units f ∈ B[[t]]× such that

(1) f(0) = 1,(2) f(t1 + t2) = f(t1)f(t2) for free commutative variables t1, t2,

with B-module structure given by (b · f)(t) = f(bt).

As remarked in [BO78, p. 1], the following property of ΓA is “in a way, a multiplicativeversion of SymA” as SymA is the left-adjoint of the forgetful functor from commutativeA-algebras to A-modules.

Proposition 1.1.3.7 ([BO78, Proposition A1], [Rob63, Theorem III.1]). For B a com-mutative A-algebra, there is a canonical bijection

(1.1.3.8) HomAlgA(ΓA(M), B)∼−→ HomA−mod(M, exp(B))

9

given by the relation, for f ∈ HomAlgA(ΓA(M), B), g ∈ HomA−mod(M, exp(B)), and m ∈M ,

(1.1.3.9) g(m) =∞∑n=0

f(m[n])tn,

i.e. ΓA is left-adjoint to exp.

Proof. Write GA(M) for the free polynomial algebra over A generated by the symbolsm[i] for m ∈ M and i ≥ 0 and IA(M) for the ideal of relations of conditions (1) to (4) of

Definition 1.1.3.1, so that GA(M)/IA(M)∼→ ΓA(M).

Given any map of sets g : M → B[[t]], we have coefficient functions bg = b : M × N→ Bso that g(m) =

∑∞i=0 b(m, i)t

i. We observe that this defines a map b : GA(M) → B, andthat the associations g 7→ bg 7→ b ∈ HomA(GA(M), B) are bijective. All that we need to dois to show that b(IA(M)) = 0 if and only if the image of g lies in exp(B) ⊂ B[[t]] and g is amap of A-modules.

We will progress through the generators of IA(M) given by conditions (1) to (4) ofDefinition 1.1.3.1 in sequence.

(1) We see that b kills (m, 0) − 1 for all m ∈ M if and only if the leading coefficientof g(m) is 1 for all m ∈ M , i.e. g(m)(0) = 1, which is condition (1) of Definition1.1.3.6 that g(m) must satisfy in order that g(m) ∈ exp(B) ⊂ B[[t]].

(2) We see that b kills (am, i) − ai(m, i) for all m ∈ M, i ≥ 0, a ∈ A if and only ifg(m)(at) = g(am)(t); by the A-module structure on exp(B), this means that gsatisfies the a · g(m) = g(am) condition on morphisms of A-modules.

(3) We see that b kills

(m, i)(m, j)− (i+ j)!

i!j!(m, i+ j)

for all m ∈M, i, j ≥ 0 if and only if

g(m)(t1 + t2) =∞∑k=0

b(m, k)(t1 + t2)k

=∞∑k=0

∑i+j=k

(b(m, i)b(m, j)

i!j!

(i+ j)!

)(i+ j

i

)ti1t

j2

=

(∞∑i=0

b(m, i)ti1

)(∞∑j=0

b(m, j)tj2

)= g(m)(t1) · g(m)(t2),

which is condition (2) of Definition 1.1.3.6 for g(m) to lie in exp(B) ⊂ B[[t]].

10

(4) We see that b kills (m+m′, i)−∑

p+q=i(m, p)(m′, q) for all m,m′ ∈M , i ≥ 0 if and

only if

g(m) · g(m′) =

(∞∑i=0

b(m, i)ti

)(∞∑j=0

b(m′, j)tj

)

=∞∑k=0

∑i+j=k

b(m, i)b(m′, j)ti+j

=∞∑k=0

b(m+m′, k)tk

= g(m+m′),

as required for g : M → exp(B) to obey the property g(m+m′) = g(m) + g(m′) ofA-module homomorphisms.

The following corollaries will be very useful.

Corollary 1.1.3.10 ([BO78, Proposition A2], [Rob63, Theorems III.3 and III.4]). LetA be a commutative ring.

(1) If B is a commutative A-algebra and M is an A-module, ΓA(M)⊗AB∼−→ ΓB(M⊗A

B), by sendingm[i] ⊗ 1 7→ (m⊗ 1)[i].

(2) If M = lim−→λMλ is a colimit of A-modules, then

lim−→ΓA(Mλ) ∼= ΓA(lim−→Mλ).

(3) If M1,M2 are A-modules, then there is a canonical isomorphism

ΓA(M1 ⊕M2)∼−→ ΓA(M1)⊗A ΓA(M2)

(m1,m2)[i] 7→∑p+q=i

m[p]1 m

[q]2

such that the grading on the left corresponds to the “sum” of the bi-grading on theright, i.e. there is an induced isomorphism of A-modules

ΓdA(M1 ⊕M2)∼−→

⊕d1+d2=d

Γd1A (M1)⊗A Γd2

A (M2)

We supply a proof along the lines of [BO78].

Proof. To show (1) we simply note that (1.1.3.8) can be composed with the standardcanonical isomorphisms

HomB−alg(ΓA(M)⊗A B,B)∼−→ HomAlgA(ΓA(M), B),

HomA−mod(M, exp(B))∼−→ HomB−mod(M ⊗A B, exp(B))

to establish (1).Since we can state Proposition 1.1.3.7 by saying that ΓA : A−mod −→ AlgA is left-adjoint

to exp : AlgA −→ A−mod and left-adjoint functors preserve colimits, we have (2).As ⊗A is the coproduct in the category of commutative A-algebras and ⊕ is the coproduct

in the category of A-modules, we directly derive (3) from Proposition 1.1.3.7. The explicitform of the map may be deduced from relation (4) of Definition 1.1.3.1.

11

Now we can concretely describe ΓdA(M) in the case that M is a free A-module.

Corollary 1.1.3.11 ([BO78, Proposition A3], compare [Rob63, Theorem IV.2]). If the

A-module M is free with basis eii∈I , then for d ≥ 0, ΓdA(M) is free with basis ∏

i∈I e[ki]i |∑

I ki = d, where the ki are non-negative integers.

Proof. Corollary 1.1.3.10(2) allows us to confine ourselves to finitely generated A-modules M . Corollary 1.1.3.10(3) allows us to reduce to the case that M is free of rank1! Finally, Corollary 1.1.3.10(1) allows us to reduce to the case that A = Z. If we write efor the basis of M , then the definition of the divided power algebra and its grading showthat e[d] is a generator for ΓdZ(M). Therefore it will suffice to show that for all non-negativeintegers a, a · e[d] 6= 0.

Now clearly the Taylor expansion exp(t) ∈ Q[[t]] of et lies in exp(Q). As M is free,there exists a map M → exp(Q) sending x 7→ exp(t). Then by Proposition 1.1.3.7, thereis a canonical map ΓZ(M) → Q sending e[d] to the coefficient 1/d! of td in exp(t). Clearlya/d! 6= 0 when a 6= 0, so a · e[d] 6= 0 as well.

Now we add a second tool toward proving Theorem 1.1.3.4 in addition to the exp functor:the derivative operators.

Definition/Lemma 1.1.3.12. Let M,N be A-modules, let m ∈M , and i ≥ 0. Then wedefine the derivative operator ∂im on PA(M,N) as the A-module endomorphism of PA(M,N)given by following notions.

(1) For P ∈ PA(M,N), a commutative A-algebra B, and an element x ∈M ⊗AB, thenthen Taylor expansion of P at x with respect to m is

Sm(P )B(x) := PB[t](m⊗ t+ x).

(2) We observe that the Sm(P )B are functorial in B, and thereby defines a polynomiallaw

Sm(P ) ∈ PA(M,N ⊗A A[t]).

(3) Decompose Sm(P ) into coefficient polynomial laws ∂im(P ) such that for all commu-tative A-algebras B and all x ∈M ⊗A B,

Sm(P )B(x) =∞∑i=0

∂im(P )B(x)ti

(4) For m ∈ M and i ≥ 0, write ∂im for the (A-linear) endomorphism of PA(M,N)defined by the association P 7→ ∂im(P ), and write

Sm :=∞∑i=0

∂imti

for the resulting formal power series with coefficients in EndA(PA(M,N)).(5) The A-subalgebra D ⊂ EndA(PA(M,N)) generated by ∂im | m ∈ M, i ≥ 0 is

commutative.

12

Proof. For (2) we observe that for any morphism of A-algebras B → B′, any x ∈M ⊗A B, the diagram

x //

PB[t](m⊗ t+ x)

x⊗B 1B′ // PB′[t](m⊗ t+ x⊗ 1)

commutes.For (3) we observe that the composition of the polynomial law Sm(P ) with the linear

polynomial law induced by the homomorphism of A-modules N ⊗A A[t]→ N,∑

i niti 7→ ni,

remains a polynomial law. This composition is the coefficient polynomial law ∂im(P ).For (4) we must show that ∂im : PA(M,N)→ PA(M,N) is A-linear. This is straightfor-

ward:

Sm(P + P ′)B(x) := (P + P ′)B[t](m⊗ t+ x) = PB[t](m⊗ t+ x) + P ′B[t](m⊗ t+ x)

= Sm(P )B(x) + Sm(P ′)B(x),

and Sm(a · P ) = a · Sm(P ) for a ∈ A follows similarly.The remaining claim is (5), that the operators

∂im | m ∈M, i ≥ 0 ⊂ EndA(PA(M,N))

commute. We deduce this by composing the operation Sm with itself, repeating the argumentof [BO78]: for B ∈ AlgA, m1,m2 ∈M,x ∈M ⊗A B, and indeterminates t1, t2,

PB[t1,t2](m1t1 +m2t2 + x) = PB[t1,t2](m2t2 +m1t1 + x)

Sm1(P )B[t2](m2t2 + x) = Sm2(P )B[t1](m1t1 + x)

Sm2(Sm1(P ))B(x) = Sm1(Sm2(P ))B(x),

so comparing terms of the polynomial coefficients in t1, t2, we find that

∂im2∂jm1

ti2tj1 = ∂jm1

∂im2tj1t

i2

for all i, j ∈ N.

The key role that these derivatives will play in showing that ΓdA(M) represents the functorPdA(M,−) starts to become apparent with this

Lemma 1.1.3.13. The map S := S(·) : M → D[[t]] defines an A-linear map

S : M −→ exp(D).

Proof. First we show that the image of S(·) =∑∞

i=0 ∂i(·)t

i lies in exp(D) ⊂ D[[t]]. Since

the coefficient of t0 in Sm(P )B(x) = PB[t](m⊗t+x) is given by ∂0m(P )B(x) = PB[t](x) = PB(x)

for x ∈M ⊗A B, we see that ∂0m(P ) = P , i.e. the coefficient is 1 ∈ D as desired.

The remaining condition to verify in order to see that Sm ∈ exp(D) ⊂ D[[t]] is that

Sm(t1 + t2) = Sm(t1) · Sm(t2) ∈ D[[t1, t2]].

We now write Sm(ti) in place of Sm to specify the variable put in the place of t in the originaldefinition of S. We check that the required identity is satisfied by calculating that for all

13

m ∈M,P ∈ PA(M,N), B ∈ AlgA, x ∈M ⊗A B, we have

Sm(t1 + t2)(P )B(x) = PB[t1,t2](m⊗ t1 +m⊗ t2 + x)

= Sm(t1)(P )B[t2](m⊗ t2 + x)

=(Sm(t1)Sm(t2)

)(P )B(x),

so Sm(t1 + t2) = Sm(t1) · Sm(t2) as desired.It remains to show that S(·) is a homomorphism of A-modules. This is a simple calcula-

tion: for m1,m2 ∈M and all P,B, and x as above,

Sm1+m2(P )B(x) = PB[t](m1 ⊗ t+m2 ⊗ t+ x)

= Sm1(P )B[t](m2 ⊗ t+ x)

= Sm1(Sm2(P ))B(x).

This is commutative also, and we have Sm1+m2 = Sm1 · Sm2 = Sm2 · Sm1 . Finally, fora ∈ A,m ∈M , and P,B, x as above,

Sam(t)(P )B(x) = PB[t](am⊗ t+ x)

= PB[t](m⊗ (at) + x)

= Sm(at)(P )B(x),

so Sam(t) = Sm(at) = a · Sm(t) as desired.

Now we prove Theorem 1.1.3.4, which we recall here. Given an A-module M and d ≥ 0,we have constructed the universal homogenous degree d polynomial law

Ld : M → ΓdA(M),

and want to prove that it deserves its name, i.e. for any A-module N , the natural map

HomA(ΓdA(M), N)(1.1.3.5)−→ PdA(M,N) given by composing with Ld is bijective.

Proof. (Theorem 1.1.3.4) To show the injectivity of (1.1.3.5), choosef ∈ HomA(ΓdA(M), N) and let P = f Ld. This is a homogenous degree d polynomiallaw P : M → N . For α ∈ Idd recall the coefficients P [α] of P (Definition 1.1.2.14). Form1, . . . ,md ∈M , we have, by definition of the P [α],

PA[t1,...,td](m1 ⊗ t1 + · · ·+md ⊗ td) =∑α∈Idd

P [α](m1, . . . ,md)tα.

On the other hand, because ΓdA[t1,...,td](M⊗AA[t1, . . . , td]) ∼= ΓdA(M)⊗AA[t1, . . . , td] (Corol-

lary 1.1.3.10(1)), we see that

LdA[t1,...,td](m1 ⊗ t1 + · · ·+md ⊗ td) = (m1 ⊗ t1 + · · ·+md ⊗ td)[d]

=∑α∈Idd

∏1≤i≤d

m[αi]i tαii .

Comparing coefficients, this shows that for each α ∈ Idd ,

f(m[α1]1 · · ·m[αd]

d ) = P [α](m1, . . . ,md), ∀(m1, . . . ,md) ∈Md.

As ∏d

1 m[αi]i | α ∈ Idd , (mi) ∈Md spans ΓdA(M) as a module by its construction (Definition

1.1.3.1), this shows that P = f Ld determines f , i.e. (1.1.3.5) is injective.

14

Now we show that (1.1.3.5) is surjective. Let P ∈ PdA(M,N). We need to produce alinear map f : ΓdA(M)→ N such that

f :d∏i=1

m[αi]i 7→ P [α](m1, . . . ,md), ∀(mi) ∈Md, α ∈ Idd .

For brevity we write m for (mi) = (m1, . . . ,md) and m[α] for∏d

i=1 m[αi]i .

We have S : M → exp(D) where D ⊂ EndA(PA(M,N)) is a commutative subalgebra(Definition/Lemma 1.1.3.12(5)), and therefore by Proposition 1.1.3.7, an induced homo-morphism of commutative A-algebras S : ΓA(M) → D. One can verify from the relation(1.1.3.9) that just as Sm =

∑∞0 ∂imt

i, so also S(m[i]) = ∂im. Therefore, for any m[α] ∈ ΓA(M),

S(m[α]) = ∂αm :=∏

i ∂αimi

.

Apply these constructions to P ∈ PdA(M,N) by “evaluating at zero” the derivative of Pby some ∂ ∈ D, so that for each such P we have an A-linear map

evP : D → N

∂ 7→ ∂(P )A(0)

Composing evP with S, we have an A-linear map f : ΓA(M)→ N , mapping

f : m[α] 7→ ∂αm(P )A(0).

We find these quantities as coefficients of tα by expanding the definition of the Taylor series:if we restrict α to have cardinality n, then for all B and x ∈M ⊗A B,∑

α

∂αm(P )B(x)tα = PB[t1,...,tn](m1 ⊗ t1 + · · ·+mn ⊗ tn + x),

and specializing to x = 0 ∈M , we have∑α

∂αm(P )A(0)tα = PA[t1,...,tn](m1 ⊗ t1 + · · ·+mn ⊗ tn).

The coefficient of tα is f(m[α]) ∈ N in this series, but it is also equal to P [α](m1, . . . ,mn) byDefinition 1.1.2.14. This is what we wanted to prove.

We derive a useful corollary of Theorem 1.1.3.4 regarding the functorial behavior of ΓdA.

Corollary 1.1.3.14 ([BO78, Corollary A6]). Let

M ′ ⇒M −→M ′′ −→ 0

be an exact sequence of A-modules. For d ≥ 1 and an A-module N , this induces an exactsequence of modules of A-polynomial laws

0 −→ PdA(M ′′, N) −→ PdA(M,N) ⇒ PdA(M ′, N),

and an exact sequence

ΓdA(M ′) ⇒ ΓdA(M) −→ ΓdA(M ′′) −→ 0.

Proof. For any commutative A-algebra B,

M ′ ⊗A B ⇒M ⊗A B −→M ′′ ⊗A B −→ 0

is also exact. This is what we need in order to see the first exact sequence. The second exactsequence then follows from Theorem 1.1.3.4.

15

1.1.4. Symmetric Tensor Algebras. In the case that M is a free A-module, thereis an isomorphism of A-modules between the symmetric tensors of M of degree d and thedth graded component of the divided power algebra for M . This will be helpful later inunderstanding the multiplication law on the algebra representing the functor of pseudorep-resentations.

First we define the A-algebra of symmetric tensors.

Definition 1.1.4.1 ([Rob63, §III.5]). Let TSdA(M) be the submodule of M⊗d of elementsinvariant under the natural action of the symmetric group Sd. Let TSA(M) := ⊕d≥0TSdA(M)

with the following structure of a graded algebra. For x ∈ TSdA(M), x′ ∈ TSd′

A(M), let

D = d + d′ and consider x ⊗ x′ ∈ M⊗DA . Then x ⊗ x′ is invariant under the subgroupSd×Sd′ → SD, where the injections are defined by the ordering of the coordinates of x⊗x′.Let (σ) be a set of representatives of the left cosets of Sd × Sd′ . Define multiplication∗ : TSA(M)× TSA(M)→ TSA(M) by extending the multiplication on monomials

(1.1.4.2) x ∗ x′ :=∑σ

σ(x⊗ x′).

This multiplication is clearly bilinear so factors through TSA(M)⊗ATS(M), but showingthat it is associative takes a bit more work. For a full presentation of the commutativity andassociativity, see [Rob63, §III.5].

We observe that the map

M −→M⊗d

m 7→ m⊗ · · · ⊗m =: m⊗dA

is compatible with ⊗AB for commutative A-algebras B and therefore defines a polynomiallaw that is homogenous of degree d. Therefore, by Theorem 1.1.3.4, we have a canonicalmap

(1.1.4.3) ΓdA(M) −→ TSdA(M).

This map is characterized by the property that m[d] 7→ m⊗dA for all m ∈ M [Rob63, Propo-

sition III.1]. Using this property, one can see that the relations of Definition 1.1.3.1 definingΓA(M) as a quotient of the free commutative algebra on m[i] (m ∈ M, i ≥ 0) are sent to

zero under the map from this free commutative algebra to TS(M) defined by m[i] 7→ m⊗iA ,

so that we have a canonical map

(1.1.4.4) ΓA(M) −→ TSA(M).

See [Rob63, Proposition III.1] for further detail.This map is often an isomorphism!

Proposition 1.1.4.5 ([Rob63, Proposition IV.5]). When M is either free or is projectiveof finite rank as an A-module, then (1.1.4.4) is an isomorphism of A-modules, induced by anisomorphism of graded A-algebras

ΓA(M)∼−→ TSA(M).

Proof. When M is projective of finite rank, we reduce to the case that M is free, as allof the arguments below commute with localization.

Let M be a free A-module and choose a basis (ei)i∈I . Choose a total ordering on the

index set I for the basis. Consider the set of simple monomials ei1⊗ . . .⊗ eid ∈M⊗d and the

16

equivalence classes under the action of Sd. The ordering gives us a unique representative ofeach class with the property that

i1 ≤ i2 ≤ · · · ≤ id.

Let K be the set of equivalence classes, and for K ∈ K let eK represent the sum of theelements of the equivalence class and let eK represent the unique representative specifiedabove of the class K. This representative eK may be uniquely written (with a new choice ofindices ij ∈ I) in the form

(1.1.4.6) eK = e⊗k1

i1⊗ · · · ⊗ e⊗khih

where i1 < i2 < · · · < ih,

h∑j=1

kj = d.

We note thatk1! · · · kh! · eK =

∑σ∈Sd

σ(eK).

Choose some x ∈ TSd(M), which may be uniquely written as

x =∑

i1,...,id∈I

λi1,...,idei1 ⊗ . . .⊗ eid .

Because this is a symmetric tensor, for all σ ∈ Sd we have λiσ(1),...,iσ(d)= λi1,...,id . This means

that we can write x uniquely as

x =∑K∈K

λKeK ,

where we can set λK = λi1,...,id for any (i1, . . . , id) such that ei1 ⊗ . . . ,⊗eid ∈ K. This showsthat eKK∈K is a basis for TSdA(M). Now we will show that this basis is the image of abasis for ΓdA(M).

We recall from Corollary 1.1.3.11 that ΓdA(M) is free with the set

(1.1.4.7) h∏j=1

e[kj ]ij| conditions of (1.1.4.6) on ij, kj ⊂ ΓdA(M)

being a basis over A. This basis is in natural bijective correspondence with K. We will bedone if we can show that the map ΓdA(M)→ TSdA(M) preserves the correspondence between

their respective bases and K. Because m[i] 7→ m⊗i

for all m ∈ M, i ≥ 0, the image of

e[k1]i1· · · e[kh]

ih∈ ΓdA(M) is

e⊗k1

i1∗ · · · ∗ e⊗khih

.

Using (1.1.4.2), we find that this product is precisely the sum over the permutations of the

orderings of factors e⊗kj

ij, which is the basis element eK

1.1.5. Faithful Polynomial Laws. We introduce the notion of a kernel of a polynomiallaw P : M → N , which is the kernel of the surjection from M onto the smallest quotientA-module of M through which a polynomial law P ∈ PA(M,N) factors.

Definition 1.1.5.1 ([Che11, §1.17]). Let P ∈ P (M,N). Then ker(P ) ⊂M is the subsetof elements m ∈M such that

∀B ∈ AlgA,∀b ∈ B, ∀x ∈M ⊗A B, P (m⊗ b+ x) = P (x),

17

which we immediately observe is a A-submodule of M . By Proposition 1.1.2.16, m is inker(P ) if and only if for all n ≥ 1 and m1, . . . ,mn ∈M ,

P (tm+ t1m1 + · · ·+ tnmn) ∈ N [t, t1, . . . , tn]

lies in N ⊗A A[t1, . . . , tn], i.e. it is independent of t.When ker(P ) = 0, we say that P is faithful.

This lemma shows that the kernel deserves its name.

Lemma 1.1.5.2 ([Che11, Lemma 1.18]). Let P ∈ PA(M,N).

(1) ker(P ) is the biggest A-submodule K ⊂ M such that P admits a factorization P =

P π where π is the canonical A-linear surjection M −→M/K.

(2) P : M/ ker(P ) −→ N is a faithful polynomial law, and if P is homogenous of degree

d, so is P .(3) If B is a commutative A-algebra, then the image of

ker(P )⊗A B −→M ⊗A Bis contained in ker(P ⊗A B).

Proof. Assertion (3) follows from transitivity of the tensor product − ⊗A B ⊗B Cinvolved in the “restriction” of P to B-algebras C through the morphism A→ B.

Clearly if P factors through M → M/K for some A-submodule K of M , then K is inthe kernel of P by definition. Now we check the converse: say K ⊂ ker(P ). We need to

factor P through a polynomial law P : M/K −→ N .For any commutative A-algebra B, set

KB := =(K ⊗A B −→M ⊗A B).

Then ι : (M/K) ⊗A B∼→ (M ⊗A B)/KB by the right-exactness of the tensor product

functor − ⊗A B, and KB ⊂ ker(P ⊗A B) by part (3). Now, applying the assumption thatK ⊂ ker(P ) and the definition of the kernel, we observe that the map PB : M⊗AB → N⊗ABsatisfies PB(k + m) = PB(m) for any m ∈ M ⊗A B, k ∈ KB, so that PB factors throughπB : M ⊗A B → (M ⊗A B)/KB. Composing this map with the isomorphism above, we havea map which is well-defined by the relation

(1.1.5.3)PB : (M/K)⊗A B → N ⊗A B

PB(ι−1 πB(M)) := PB(m)

because ι−1 πB is surjective onto (M/K) ⊗A B. As all of the maps defining this relation

are functorial in B, we have defined an polynomial law P ∈ PA(M/K,N).

Because of the relation (1.1.5.3), we see that ker(P ) = ker(P )/K. This is the first partof (2), and the second part of (2) also follows from examining (1.1.5.3).

1.1.6. Multiplicative Polynomial Laws. Now we consider polynomial laws betweenA-algebras. These are polynomial laws between the underlying A-modules with multiplica-tivity imposed. It is possible to define these laws when A is neither associative nor unital.

Definition 1.1.6.1. Let R, S have the structure of A-algebras (not necessarily commu-tative) and P ∈ PdA(R, S). Then P is called multiplicative provided that

(1) P (1) = 1, i.e. PA(1R) = 1S, from which it follows that PB(1R⊗AB) = 1S⊗AB for allcommutative A-algebras B, and

18

(2) PB is multiplicative for all commutative A-algebras B, i.e. for all B and all x, y ∈R⊗A B, PB(xy) = PB(x)PB(y).

For d ≥ 0, we denote byMdA(R, S) ⊂ PdA(R, S) the set of degree d homogenous multiplicative

polynomial laws from R to S over A.

Remark 1.1.6.2. A multiplicative polynomial law of degree 0 must be constant, and somust send every element to the multiplicative identity.

We now ask if there exists a universal object for the functor MdA(R,−) on the category

of A-algebras. Of course, an element of MdA(R, S) induces an element of PdA(R, S) by the

forgetful functor from A-algebras to A-modules, so that P ∈MdA(R, S) induces a morphism

of A-modules ΓdA(R)→ S. The composite

R→ ΓdA(R)→ S

is multiplicative. In fact, there exists a A-algebra structure on ΓdA(R) such that the firstmap of the composite is multiplicative, and the multiplicativity of the composite dependson the multiplicativity of the second map. This reasoning, due to Roby [Rob80], makes thefirst map a universal homogenous degree d multiplicative polynomial law. This is what wewill now explain.

Let M,N ∈ A−mod, and for d ≥ 0 write LdM for the universal homogenous degreed polynomial law LdM : M → ΓdA(M). By the universal property of the tensor productof modules, M ⊗A N is universal for bilinear maps out of M × N . The universal mapM ⊕N → M ⊗A N is manifestly homogenous of degree 2 and compatible with −⊗A B forcommutative A-algebras B. Therefore we have a degree 2 homogenous polynomial law fromM ⊕N to M ⊗A N , which we will denote by βM,N ∈ P2

A(M ⊕N,M ⊗A N).The composition LdM⊗AN βM,N defines a degree 2d polynomial law in P2d

A (M ⊕N,M ⊗AN), so that by Theorem 1.1.3.4 there exists a canonical A-linear homomorphism

ηM,N : Γ2dA (M ⊕N) −→ ΓdA(M ⊗A N)

such that LdM⊗AN βM,N = ηM,N L2dM⊕N is an equality of polynomial laws. Recall from

Corollary 1.1.3.10(2) that there is a canonical isomorphism⊕p+q=2d

ΓpA(M)⊗A ΓqA(N)∼−→ Γ2d

A (M ⊕N),

and we will also write ηM,N for its restriction to ΓpA(M)⊗A ΓqA(N) for p+ q = 2d, consideredas a submodule of Γ2n

A (M ⊕N).

Sublemma 1.1.6.3 ([Rob80, p. 869]). With M,N, d, βM,N , and ηM,N as above, let

m[α] := mα11 · · ·mαr

r , wherer∑i=1

αi = p

n[α′] := nα′11 · · ·nα

′ss , where

s∑j=1

α′j = q

be representative elements of ΓpA(M) and ΓqA(N), respectively. Then

ηM,N(m[α] ⊗ n[α′]) =∑γij

∏1≤i≤r1≤j≤s

(mi ⊗ nj)[γij ],

19

where αi =∑

j γij and α′j =∑

i γij. In particular, ηM,N kills ΓpA(M)⊗A ΓqA(N) ⊂ Γ2dA (M ⊗A

N) if p 6= q.

Now we replace M and N with an A-algebra R, and write ηR (resp. βR) for ηM,N

(resp. βM,N). Let θ : R ⊗A R → R be the linear multiplication structure map for R.As ΓA is a functor, the data of θ as a map of A-modules induces a morphism of A-algebras

Γ(θ) : ΓA(R⊗A R) −→ ΓA(R),

which restricts to its graded components Γn(θ). Now write θd for the composition

θd : ΓdA(R)⊗ ΓdA(R)ηR−→ ΓdA(R⊗A R)

Γd(θ)−→ ΓdA(R).

Lemma 1.1.6.4 ([Rob80, p. 870]). The A-linear map θd defines the structure of an A-algebra on ΓdA(R). If R is

(1) unital,(2) associative, or(3) commutative,

then for all d ≥ 0, ΓdA(R) is as well.

Now we know that an associative unital A-algebra R gives rise to an associative unitalA-algebra ΓdA(R). We can also check that the universal polynomial law LdR : R → ΓdA(R) ofTheorem 1.1.3.4 is multiplicative with respect to this structure. In fact, this the multiplica-tive polynomial law LdR is universal for multiplicative homogenous degree d polynomial lawsout of R.

Theorem 1.1.6.5 ([Rob80, Theorem]). For A-algebras R, S, there is a canonical bijection

MdA(R, S)

∼−→ HomAlgA(ΓdA(R), S)

with universal object (LdR : R −→ ΓdA(R)) ∈MdA(R,ΓdA(R)).

Recall that the kernel of a polynomial law P ∈ PA(M,N) is kernel of the factor map to thesmallest quotient A-module of M through which P factors. Naturally, in the multiplicativecase, we would like this quotient to be a quotient ring and the kernel to be a two-sided idealof a multiplicative polynomial law P ∈ Md

A(R, S). The following lemma proves this andprovides a simplification of the description of ker(P ) for the multiplicative case relative tothe module-theoretic case.

Lemma 1.1.6.6 ([Che11, Lemma 1.19]). Let R, S be a A-algebras and let P ∈MdA(R, S).

Then

(1) The submodule ker(P ) ⊂ R defined in Definition 1.1.5.1 satisfies

ker(P ) = r ∈ R | ∀B, ∀r′ ∈ R⊗A B, P (1 + rr′) = 1,and the same equality holds on replacing the condition P (1 + rr′) = 1 with P (1 +r′r) = 1.

(2) ker(P ) ⊂ R is a two-sided ideal of R. It is proper if d > 0, and it is the biggest

two-sided ideal K ⊂ R such that P admits a factorization P = P π where π is the

standard surjection π : R→ R/K and P ∈MdA(R/K, S).

There is sometimes an even more concise description of the kernel, which we leave toLemma 1.1.7.2 below.

20

Proof. Write J1(P ) for the right and side of the equality in (1), and write J2(P ) forthe same set with the condition P (1 + r′r) = 1 in place of P (1 + rr′) = 1. First, weshow that ker(P ) ⊆ J1(P ). The same argument will show that ker(P ) ⊆ J2(P ). Chooser ∈ ker(P ), a commutative A-algebra B, and r′ = 1+h ∈ R⊗AB. It will suffice to show thatP (1+r(1+th)) is the unit polynomial 1 in S⊗AB[t]; we can then deduce the desired identityby specializing t to t = 1. Since this polynomial has degree at most d in t, it will suffice tocheck that this holds in S⊗AB[t]/(td+1). Notice that 1+ th is invertible in R⊗AB[t]/(td+1).Applying multiplicativity and the definition of the kernel (Definition 1.1.5.1), we have

P (1 + r(1 + th)) = P ((1 + th)−1 + r)P (1 + th) = P ((1 + th)−1)P (1 + th) = P (1) = 1.

Therefore, J1(P ) ⊆ ker(P ), and the analogous calculation with P (1 + (1 + th)r) shows thatJ2(P ) ⊆ ker(P ).

A similar argument shows that ker(P ) ⊇ J1(P ): choose B, r, and r′ = 1 + h as above.Using the fact that r ∈ J1(P ), and calculating in S ⊗A B[t]/(td+1), we have

P (rt+ (1 + h)) = P ((1 + rt) + h) = P (1 + (1 + rt)−1h)P (1 + rt)

= P (1 + (1 + rt)−1h) = P (1 + h+ r(· · · )) = P (1 + h),

and the lack of dependence on t shows that r ∈ ker(P ) by definition. The same argumentshows that ker(P ) ⊆ J2(P ) as well.

By part (1), ker(P ) is visibly a two-sided ideal of R. The rest of part (2) follows directlyfrom the calculations in the proof of Lemma 1.1.5.2, in particular (1.1.5.3).

1.1.7. Definition of Pseudorepresentations and Representability of the Pseu-dorepresentation Functor. With the background above in place, we can restate the def-inition of pseudorepresentations in terms of polynomial laws, and immediately make sev-eral conclusions based on the theory of multiplicative homogenous polynomial laws outlinedabove.

Definition 1.1.7.1 (Reprising Definition 1.1.1.1). Let A be a commutative ring, let Rbe an A-algebra, and let d ≥ 0. A d-dimensional pseudorepresentation D of R over A is adegree d homogenous multiplicative polynomial law

D : R −→ A.

The set of d-dimensional pseudorepresentations of R over A is denoted PsRdR(A). When

B is a commutative A-algebra, we use PsRR(B) to denote the set of d-dimensional pseu-dorepresentations of R⊗AB over B, and we observe that PsRd

R is naturally a functor underthe tensor product. Following Remark 1.1.6.2, we note that there is always a unique degree0 pseudorepresentation sending everything to the multiplicative identity. We will formallyset this to be the determinant of the unique “zero-dimensional representation.”

We will freely use the notions associated to multiplicative polynomial laws to describepseudorepresentations. One particular notion that we will use heavily of is the kernel of apseudorepresentation D : R → A, written ker(D). This is a two-sided ideal of R, whichis the kernel of the surjection of A-algebras R R/ ker(D) with the special property thatthis is the smallest quotient of R through which D factors (cf. Lemma 1.1.6.6). The fol-lowing lemma is special to the case that a multiplicative A-polynomial law P : R → S is apseudorepresentation (i.e. S = A).

21

Lemma 1.1.7.2. Let A be an infinite cardinality commutative domain and let D : R→ Abe a d-dimensional pseudorepresentation. Then

ker(D) = r ∈ R | ∀r′ ∈ R,D(1 + rr′) = 1.

Proof. Combine Proposition 1.1.2.11 and Lemma 1.1.6.6.

Remark 1.1.7.3. In order to call a homogenous multiplicative polynomial law a pseu-dorepresentation of R over A, it is occasionally important to be precise about the stipulationthat the target is A and the source R is an A-algebra. It is common and reasonable todepart from this precision in the following case: if B is a commutative A-algebra, a degreed homogenous multiplicative A-polynomial law

R −→ B

is not, strictly speaking, a pseudorepresentation of R into B or over B (there is no suchthing because R is not a B-algebra). However, the distinction is not vast, because this datainduces a degree d homogenous multiplicative polynomial law

R⊗A B −→ B

which is a d-dimensional pseudorepresentation of R ⊗A B over B. By Corollary 1.1.3.10,this induction of a pseudorepresentation from a multiplicative polynomial law is a bijection.Therefore, we call the degree d homogenous multiplicative polynomial law R −→ B a d-dimensional pseudorepresentation of R valued in B.

The results of Roby on homogenous multiplicative polynomial laws immediately implyimportant facts about pseudorepresentations. To state these, we recall that the abelian-ization Rab of an algebra R is its quotient by its two-sided ideal generated by xy − yx forx, y ∈ R. Obviously this quotient has the universal property expected of the abelianization.

Theorem 1.1.7.4 ([Che11, Proposition 1.6]). Let R be an A-algebra and d ≥ 1. Thefunctor PsRd

R : AlgA → Set is representable by the commutative A-algebra

ΓdA(R)ab,

with universal pseudorepresentation

Du : R −→ ΓdA(R)ab

r 7→ r[d]

Moreover, for any commutative A-algebra B,

(1) There is an isomorphism of functors on B-algebras

PsRR ×SpecA SpecB∼→ PsRR⊗AB,

corresponding to the canonical isomorphism of B-algebras

ΓdA(R)ab ⊗A B∼−→ ΓdB(R⊗A B)ab.

(2) When D : R → B is a homogenous multiplicative polynomial law (of unspecifiedand possibly non-existent degree), the degree is constant on connected componentsof SpecB. In particular, if SpecB is connected, then D is a pseudorepresentationof some dimension.

22

(3) If R is free as an A-module, then PsRR(−) is represented by the commutative A-algebra TSdA(R)ab with universal pseudorepresentation

R −→ TSdA(R)

r 7→ r⊗dA

(4) If R is the group algebra over A of the group (or monoid) Γ, then

PsRdR∼−→ Spec(TSdZ(Z[Γ]))ab ×SpecZ SpecA.

(5) If R is finite as an A-module, for example the group algebra of a finite group, thenΓdA(R)ab is finite as an A-module.

(6) A d-dimensional B-valued representation of R, i.e.

R⊗A B −→ E ,where E is a rank d2 Azumaya B-algebra, induces a pseudorepresentation by com-position with the reduced norm E → B.

For the definition of an Azumaya algebra, see Definition 1.4.1.5.Before proving this theorem, which summarizes our knowledge up to this point, we note

the most glaringly missing basic fact about the pseudorepresentation functor: we do not knowif finite generation (or some other condition other than finiteness of R as an A-module) ofR over A implies finite generation of ΓdA(R)ab. This is true (it is Theorem 1.1.10.15), butwill require the study of pseudorepresentations on freely generated (non-commutative) A-algebras in §1.1.9, and the application of invariant theory.

Remark 1.1.7.5. Recall that there is a unique 0-dimensional pseudorepresentation whichsends everything to the multiplicative identity. This corresponds to the fact that Γ0

A(R) ∼= A,i.e. PsR0

R = SpecA.

Proof. The main theorem statement follows closely from the representability statementfor homogenous multiplicative polynomial laws, Theorem 1.1.6.5. Indeed, we know that fora commutative A-algebra B, the association

MdA(R,B)

∼−→ HomAlgA(ΓdA(R), B)

is an isomorphism, and the right hand side is canonically isomorphic to HomA(ΓdA(R)ab, B)since B is commutative. As the association

MdA(R,B)

∼−→ PsRdR(B)

D 7→ D ⊗A Bdiscussed in Remark 1.1.7.3, following Corollary 1.1.3.10, is bijective, we have the theorem.In addition, part (1) follows directly from Corollary 1.1.3.10 and the main theorem.

Forgetting the algebra structure on R and B, the polynomial law D induces a map ofmodules ΓA(R) → B by Theorem 1.1.3.4. By the proof of Theorem 1.1.6.5, the multiplica-tivity of D implies that ΓA(R)→ B is multiplicative, where this multiplication operation isnot the multiplication of the divided power algebra, but the multiplication on each gradedcomponent ΓdA(R), d ≥ 0. As B is commutative, this amounts to an A-algebra homomor-phism ∏

d≥0

ΓdA(R)ab −→ B.

23

As B has no nontrivial idempotents, this map factors through one of the factors, say of degreed, so that D is a homogenous multiplicative polynomial law of degree d. This establishes(2).

Part (3) follows directly from Proposition 1.1.4.5. Part (4) follows from part (3) and thefact that a group algebra A[Γ] is equal to Z[Γ]⊗Z A. Part (5) is quickly checkable, say withthe explicit generators for ΓdA(R) given in Corollary 1.1.3.11, along with Corollary 1.1.3.14.

Let Md be the ring-scheme over SpecZ, the d-by-d matrix algebra. Each coefficientof the characteristic polynomial defines a regular function Md → A1

Z which is equivariantunder the adjoint action of PGLd on Md and the trivial action on A1. Each Azumayaalgebra E is a form of Md twisted by this action (cf. [Gro68, Corollary 5.11]); therefore,the characteristic polynomial function descends from E ⊗OX OU ∼= Md(OU) to E over OX[Gro68, 5.13]. As PsRd

R is an etale sheaf (it is representable by a scheme), the formation ofa pseudorepresentation by taking the determinant of a representation into a matrix algebradescends to the case of a representation into an Azumaya algebra. This establishes (6).

Remark 1.1.7.6. For any A-scheme X, we may extend the definition of a pseudorepre-sentation to allow for a OX-valued d-dimensional pseudorepresentation of R. This functoris still represented by the affine A-scheme PsRd

R.

Example 1.1.7.7. Let R = A[X]. Then R is free as an A-module, so by Theorem1.1.7.4(4), PsRd

R is represented by the d graded piece of the symmetric tensor algebra

TSdA(A[X])ab = A[X1, . . . , Xd]Sd = A[Σ1, . . . ,Σd],

where (Σi) are the standard symmetric polynomials on d variables. The universal pseudorep-resentation A[X] → A[Σ1, . . . ,Σd], X 7→ Σd is realized as the associated pseudorepresenta-tion of a d-dimensional representation: let A[X] act on the rank d free A[Σ1, . . . ,Σd]-module

A[Σ1, . . . ,Σd][Y ]/(Y d − Σ1Yd−1 + . . .+ (−1)dΣd)

by X 7→ Y . The characteristic polynomial of X is the standard one, i.e. the generator of theideal in the line above.

Example 1.1.7.8. Let A = Z and let Γ = Z. Letting X represent a generator of Γ, wewrite R = A[X,X−1]. As in the previous example, the dth graded component of the symmet-

ric tensor algebra represents PsRdR. We observe that R⊗

dZ is a standard presentation of the

coordinate ring of the split rank d torus Gdm/ SpecZ, and that its subring TSdZ(Z[X,X−1]) is

the subring of invariants of the action of the Weyl group. Via the Chevalley isomorphism, thegeometric points of Spec TSdZ(R) are in natural bijective correspondence with the semisimplegeometric points of GLd, up to conjugation. This latter set is clearly in natural bijectivecorrespondence with d-dimensional semisimple representations of Z up to isomorphism.

Example 1.1.7.9. As noted in the theorem, for a finite group Γ, TSdZ(Z[Γ]) is a finite

Z-module. We observe that Z[Γ]⊗d ∼= Z[Γ×

d] is generated (as a module, even) by elements of

finite multiplicative order, i.e. γn = id for some n ≥ 1. Therefore its subquotient TSdZ(Z[Γ])ab

consists of sums with coefficients in Z of elements of finite multiplicative order.Now fix a d-dimensional complex representation of Γ. By the discussion of §1.1.1,

we may associate to this representation a C-valued pseudorepresentation, and therefore amap TSdZ(Z[Γ])ab → C. The property above shows that the image must be contained inlim−→n

Z[µn] ⊂ C, where µn is a primitive nth root of unity. Since the image of the map is

24

generated by characteristic polynomial coefficients of elements of Γ, we observe the well-known fact that characteristic polynomial coefficients of a given representation generate acyclotomic integral extension of Z.

Such observations also may be used to observe the Brauer character theory of positivecharacteristic representations of Γ.

We now investigate the pseudorepresentations of a rank n2 Azumaya algebra R over acommutative ring A. This includes the case that R = Mn(A), as Azumaya algebras are etalelocally matrix algebras (see Definition 1.4.1.5). Since the representation theory of a matrixalgebra over a field consists of direct sums of the identity representations, we expect this tobe reflected in its pseudorepresentations. This is what we record in the following proposition,due to Ziplies [Zip86].

Proposition 1.1.7.10 ([Zip86], see also [Che11, Exercise 2.5]). Let R be an Azumaya A-algebra of rank n2. Then the pseudorepresentations D : R→ A consist precisely of powers ofthe reduced norm detR : R→ A. In other words, the reduced norm induces an isomorphism,for each d ≥ 0 divisible by n,

ΓdA(R)ab ∼−→ Γd/nA (A) ∼= A,

and for n - d there are no d-dimensional pseudorepresentations of R.

Remark 1.1.7.11. Proposition 1.1.7.10 reflects the fact that the basic algebra corre-sponding to e.g. Md(C) is C, and that Md(C) and C are Morita equivalent. See Definition2.2.2.1 for the notion of a basic algebra, see Definition 2.2.2.8 for the notion of a basic al-gebra associated to an algebra, and Theorem 2.2.2.10 for the fact that their representationcategories are equivalent.

We record for future reference some important qualities of the functor ab sending A-algebras to commutative A-algebras.

Lemma 1.1.7.12 (cf. [Vac08, Lemma 5.14]). If f : R S is a surjection of A-algebras,then

(1) The induced homomorphism of commutative A-algebras f ab : Rab → Sab is alsosurjective, and

(2) ker f ab ∼= abA(ker f), where abA : A Aab is the canonical homomorphism.

Proof. Let Iab(A) be the kernel of abA. Then the lemma follows from the snake lemmaapplied to the commutative diagram

Iab(A) //

Iab(B) //

0

0 // ker f //

abA

Af //

abA

B //

abB

0

0 // ker f ab // Aab fab

//

Bab //

0

0 0

25

1.1.8. The Characteristic Polynomial. Let R be an A-algebra and A be a com-mutative ring as usual, and fix for this subsection a d-dimensional pseudorepresentationD : R → A. As described in the introduction §1.1.1, D induces a characteristic polyno-mial function χD(·, t) : R → A[t] according to the following definition, repeated from theintroduction.

Definition 1.1.8.1. Given a d-dimensional pseudorepresentation D : R −→ A of R overA, its characteristic polynomial function is

χD(·, t) : R −→ A[t]

r 7→ DA[t](t− r) = td − Λ1(r)td−1 + · · ·+ (−1)dΛd(r)

where Λi are maps R→ A.

In fact, by taking B-valued pseudorepresentaitons of R for commutative A-algebras B,the characteristic polynomial coefficients are homogenous polynomial laws extending thefunctions Λi : R→ A described above.

Definition/Lemma 1.1.8.2. With D,R,A as above and a commutative A-algebra Band i ≥ 0, let the function Λi,B : R⊗A B → B be defined by the formula, for r ∈ R⊗A B,

χD(r, t) := DB[t](t− r) =d∑i=0

(−1)iΛi,B(r)td−i.

is a homogenous A-polynomial law of degree i from R to A. We also have characteristicpolynomial coefficient polynomial laws Λi, where Λ0 ≡ 1 and Λd = D are multiplicative, andfor i ≥ d+ 1 we set Λi ≡ 0. We call Λ1 the trace.

We note that the data of the polynomial law Λ1 is characterized by the A-linear mapR→ A (cf. Example 1.1.2.8).

Proof. We must prove the implied claim that Λi is a homogenous polynomial law ofdegree i. First we observe that Λi is a polynomial law, since the formula for Λi,B given acommutative A-algebra B can be checked to be functorial in B. For b ∈ B and x ∈ R⊗AB,Λi,B(bx) is the coefficient of td−i in DB[t](t − bx). If we write t = t1 and let t2 be anindeterminant, then the functorality of polynomial laws shows that ΛD

i is the specializationof DB[t1,t2](t1 + t2(−x)) via B[t1, t2]→ B[t], t1 7→ t, t2 7→ b. By Proposition 1.1.2.16, the onlynonzero coefficient of DB[t1,t2] where t1 appears to the (d− i)th power also has t2 to the ithpower. Therefore Λi,B(bx) = bi · Λi,B(x) as desired.

The characteristic polynomial coefficient polynomial laws allow for another descriptionof the kernel of D : R→ A:

ker(D) = r ∈ R | ∀B, ∀r′ ∈ R⊗A B, ∀i ≥ 1,Λi,B(rr′) = 0One can, therefore, give a description of the kernel of D as the set of elements of R suchthat any multiple of r in R ⊗A B for all B has the characteristic polynomial td. When A isan infinite domain, this criteria still works when applied only to R (see Lemma 1.1.7.2).

Example 1.1.8.3. Let Td(A) ⊂Md(A) be the A-subalgebra of upper triangular matrices,with the pseudorepresentation D : Td(A) → A induced from the determinant on Md(A).Then ker(D) is the ideal of strictly upper triangular matrices, and D factors through thediagonal subalgebra Td(A)/ ker(D) of Md(A).

26

Implicit in the example above is the Cayley-Hamilton theorem: When R is a matrixalgebra Md(A) and D = det is induced by the standard determinant, this defines a degree dpolynomial law, and the characteristic polynomial χD is the same as the standard character-istic polynomial. It is very important that each element r ∈ R satisfies its own characteristicpolynomial! That is, χ(r, r) = 0. This is the Cayley-Hamilton theorem. For a generalA,R,D, this may not be the case, and the following polynomial laws measure this failure.

Definition 1.1.8.4. With A,R,D, and χ as above, let χ : R −→ R be the homogenousdegree d A-polynomial law

χ(r) = χD(r, r) = rd − Λ1(r)rd−1 + Λ2(r)rd−2 + · · ·+ (−1)dΛd(r).

For any n ≥ 1 and α ∈ Inn (the set of n-tuples of non-negative integers (α1, . . . , αn) with sumn), we recall that χ[α] of Definition 1.1.2.14 are the coefficient functions of χ, defined by therelation

χR[t1,...,tn](r1t1 + · · ·+ rntn) =∑α∈Inn

χ[α](r1, . . . , rn)tα,

where tα =∏n

1 tαii .

We recall Proposition 1.1.2.16(3): because χ is homogenous of degree d, χ[α] 6≡ 0 onlywhen n = d, and χ is characterized by the functions χ[α] | α ∈ Idd. Therefore givenA,R,D, d as usual, every element r ∈ R “satisfies its characteristic polynomial” if and onlyif χ ≡ 0 as a polynomial law if and only if χ[α] ≡ 0 for all α ∈ Idd .

This equivalence results in the notion of a Cayley-Hamilton pseudorepresentation.

Definition 1.1.8.5 (cf. [Che11, p. 17]). Let R be an A-algebra and let D be a d-dimension pseudorepresentation. Let CH(D) ⊂ R be the two-sided ideal generated byχ[α](r1, . . . , rd) as (ri) varies over all d-tuples in R and α varies over Idd . We say that Dis Cayley-Hamilton if CH(D) = 0. Equivalently, χ ≡ 0 as a polynomial law. We also saythat (R,D) is a Cayley-Hamilton A-algebra of degree d.

Of course, R/CH(D) is a Cayley-Hamilton A-algebra.The following observation will be very important in the sequel (see e.g. Proposition

1.2.4.3).

Lemma 1.1.8.6. The Cayley-Hamilton property of a pseudorepresentation D : R→ A isstable under base changes ⊗AB, i.e. if (R,D) is a Cayley-Hamilton A-algebra, then (R ⊗AB,D⊗AB) is as well. In particular, if D is an arbitrary d-dimensional pseudorepresentation,then there is a natural isomorphism

(1.1.8.7) R/CH(D)⊗A B∼−→ (R⊗A B)/CH(D ⊗A B).

Proof. The Cayley-Hamilton property and the Cayley-Hamilton ideal CH(D) are func-torial under base change because they are defined by the image of the A-polynomial lawχ : R → R, and the functions χB as B varies over the category of commutative A-algebrasis functorial (1.1.2.2). Therefore, if (R,D) is Cayley-Hamilton, then χ = χD is equal to 0as an A-polynomial law; therefore χB, being simply a restriction of χ from the category ofA-algebras to the category of B-algebras via the map A→ B, is still 0. This proves the firstpart of the statement of the lemma.

By the definition of CH(D) as the ideal of R generated by the images of an A-polynomiallaw, we see that there exists a map

(1.1.8.8) CH(D)⊗A B → CH(D ⊗A B) ⊂ R⊗A B

27

and that this map is an surjection. This proves that the A-homomorphism (1.1.8.7) exists and

is injective. Using the canonical isomorphism R/CH(D)⊗A B∼→ (R⊗A B)/(CH(D)⊗A B),

we see that (1.1.8.7) is surjective, completing the proof.

1.1.9. Universal Polynomial Identities. We remarked in the introduction that apseudorepresentation of an A-algebra R over A amounts to the data of a characteristicpolynomial for each element of r. We will substantiate this comment in Corollary 1.1.9.15,showing that the characteristic polynomial coefficient functions Λi characterize a pseudorep-resentation. Conversely, given a characteristic polynomial function χ(·, t) : R → A[t], onewould have to impose a great deal of identities upon this function in order to “call it apseudorepresentation.” While we will not find a complete list of identities, in this sectionwe will prove that the characteristic polynomial of a pseudorepresentation “satisfies all ofthe identities that one would expect form the characteristic polynomials of a representation”(see (1.1.9.5)), even though it may not be induced by an actual Azumaya/matrix algebra-valued representation. After that, we will deduce a few particular, useful identities from thiscollection (Proposition 1.1.9.11).

Definition 1.1.9.1. Given a set X, the d-dimensional generic matrices representationinvolves the following data:

(1) The free Z-algebra ZX on the set X;(2) The coefficient ring FX(d) = Z[xij], the free polynomial ring on generators xij for

x ∈ X and 1 ≤ i, j ≤ d;(3) The representation

ρuniv : ZX −→Md(FX(d))

x 7→ (xij)ij

(4) We also define the subring EX(d) ⊂ FX(d) generated by characteristic polynomialcoefficients of ρuniv, i.e. by Λi,Z(x) for x ∈ ZX and for Λi,Z : ZX → FX(d) for1 ≤ i ≤ d the characteristic polynomial coefficient functions of the d-dimensionalpseudorepresentation det ρuniv : ZX −→ FX(d).

Remark 1.1.9.2. It remains to be shown that the pseudorepresentation det ρuniv factorsthrough EX(d) → FX(d). This will come along with the proof that a pseudorepresentationis determined by its characteristic polynomial coefficient functions.

Theorem 1.1.9.3 (Vaccarino [Vac08]). With notation as in Definition 1.1.9.1, the canon-ical map ΓdZ(ZX)ab → FX(d) associated to the d-dimensional pseudorepresentation

det ρuniv : ZX → FX(d)

induces a canonical isomorphism

(1.1.9.4) ΓdZ(ZX)ab ∼−→ EX(d).

We postpone to §1.1.10 the the discussion of the results of Donkin, Zubkov, and Vaccarinothat are summarized in Theorem 1.1.9.3. Here we discuss the implications of this theoremfor pseudorepresentations.

Let D : R −→ A be a d-dimensional pseudorepresentation of an A-algebra R. Let X bea set of generators for R over Z, e.g. X = R, so that there exists a surjection π : ZX R.Theorem 1.1.9.3, along with the representability Theorem 1.1.6.5, shows that there is a

28

unique ring homomorphism fX : EX(d)→ A such that

ZX det ρuniv

//

π

EX(d)

fX

RD // A

is a commutative diagram of homogenous polynomial laws over Z, where the horizontal mapshave degree d and the vertical maps have degree 1 (they are ring homomorphisms).

Using EX(d)fX−→ A −→ R where the second map is the structure map, we consider R as

an EX(d)-algebra, so that D is a homogenous multiplicative EX(d)-polynomial law of degreed. Therefore we have a diagram of homogenous multiplicative EX(d)-polynomial laws

(1.1.9.5) ZX ⊗Z EX(d)ρuniv⊗1 //

π⊗fX

Md(FX(d))det // EX(d)

fX

RD // A

As the top row factors through a matrix algebra, we can use this diagram to showthat identities in a matrix algebra, for instance, the Cayley-Hamilton identity, give rise toidentities in arbitrary homogenous multiplicative polynomial laws. One of these identities,Amitsur’s formula, requires some initial explanation.

Definition 1.1.9.6. Let X be a totally ordered finite set (alphabet), and let X+ be themonoid of words with letters in this set, with the induced total lexicographic ordering.

(1) A word w ∈ X+ is called a Lyndon word if w is less than or equal to any of itsrotations, or, equivalently, if w = xw′, then w ≤ w′. The set of Lyndon words of analphabet X is denoted LX .

(2) By the Chen-Fox-Lyndon theorem [CFL58, §1], any word w ∈ X+ may be uniquelyfactored into a Lyndon decomposition w = w1 · · ·wn, where w1 ≥ w2 ≥ · · · ≥ wn,wi ∈ LX . We also present the Lyndon decomposition as

w = wl11 wl22 · · ·wlss , where w1 > · · · > ws, wi ∈ LX .

(3) There is a unique map ε : X+ → ±1, multiplicative on Lyndon words, given bysending w to 1 if the length of its Lyndon decomposition is even, and −1 otherwise.We can write ε(w) = (−1)n or ε(w) =

∏s1(−1)li .

With the notion of Lyndon words, we can explain Amitsur’s formula.

Definition 1.1.9.7. We say that characteristic polynomial functions Λi,A : R → Asatisfy Amitsur’s formula when for any finite subset X = r1, . . . , rn ⊂ R, totally orderedby the indices, we have

(1.1.9.8) Λi,A(r1 + · · ·+ rn) =∑`(w)=i

ε(w)Λ(w),

where ` : X+ → N is the length of w in terms of the letters X, and

Λ(w) := Λls(ws) · · ·Λl2(w2)Λl1(w1).

Amitsur’s formula applies just as well to the polynomial laws Λi : R → A associatedto a d-dimensional pseudorepresentation D : R → A by applying the condition to Λi,B :

29

R⊗AB → B for every commutative A-algebra B. This gives us a notion of when D satisfiesAmitsur’s formula.

Definition 1.1.9.9 (cf. [Ami80]). For A,R, d as usual, let D : R→ B be a homogenousdegree d polynomial law into a commutative A-algebra B. Let X = r1 ⊗ t1, . . . , rn ⊗ tn ⊂R ⊗A A[t1, . . . , tn] with the standard lexicographic ordering, and preserve the notation ofDefinition 1.1.9.7 otherwise. We say that D satisfies Amitsur’s formula if

(1.1.9.10) D

(1−

n∑j=1

rjtj

)=∏w∈LX

(d∑i=0

(−1)iΛi(w)

),

where the product is taken over Lyndon words with length bounded by d, ordered decreas-ingly. Equivalently, the homogenous of degree i component of this identity holds for all1 ≤ i ≤ d:

Λi(r1t1 + · · ·+ rntn) =∑`(w)=i

ε(w)Λ(w),

where the letters in the words on the right hand side are now taken to be the n monomials“riti.”

Proposition 1.1.9.11 ([Che11, Lemma 1.12]). For A,R, d as usual, let D : R→ B be ahomogenous degree d polynomial law into a commutative A-algebra B. Let Λi,B : R→ B bethe induced characteristic polynomial coefficient polynomial laws (homogenous of degree i),and in case B = A, let χD : R → R be the degree d polynomial law given by evaluation ofthe characteristic polynomial. Then the following identities hold.

(1) (commutativity of determinant) For all r, r′ ∈ R,

D(1 + rr′) = D(1 + r′r).

(2) (Amitsur’s formula) For all r1, . . . , rn ∈ R, Amitsur’s relations (1.1.9.8) on Λi aresatisfied.

(3) (Pseudocharacter identity) The “trace function” Tr = Λ1 : R → B satisfies thed-dimensional pseudocharacter identity (1.1.12.2).

(4) (Cayley-Hamilton identity) If B = A (in which case D is a pseudorepresentation),then for all α ∈ Id, (r1, . . . , rd) ∈ Rd, and r ∈ R,

D(1 + χ[α](r1, . . . , rd) · r) = 1.

Identity (1) is basic, reflecting the fact that the characteristic polynomial coefficientsare central functions. The remaining identities have a particular, prominent use. Amitsur’sformula (2) often reduces the study of multiplicative polynomial laws to the study of theircharacteristic polynomial coefficient functions. For example, we will use it to show that thecharacteristic polynomial functions characterize a pseudorepresentation. The pseudocharac-ter identity (3) on Λ1 will allow us to compare pseudorepresentations to pseudocharacters.And the Cayley-Hamilton identity (4) will be most prominent in the new material of thisthesis, and will play a prominent role in relating pseudorepresentations to representations.

Proof. Our strategy is to use the relation (1.1.9.5) between, on the one hand, theuniversal d-dimensional pseudorepresentation induced by the determinant function Duniv =det ρuniv of the universal, generic matrices representation of the free algebra ZX withX = R, and, on the other hand, the degree d homogenous polynomial law D : R → B. Wewill show that these identities hold for the universal pseudorepresentation Duniv because it

30

is the determinant of a d-dimensional representation, and we will remark on any difficultiesin deducing the same identity for D.

We know that the characteristic polynomial functions of a representation are centralfunctions. Therefore, for r, r′ ∈ ZX ⊗Z EX(d), χDuniv(rr′, t) = χDuniv(r′r, t). Specializingto t = −1, we deduce that Duniv(1 + rr′) = Duniv(1 + r′r), proving (1).

Part (2) is precisely [Ami80, Theorem B]: the relation (1.1.9.10) is proved for the deter-minant of an arbitrary matrix algebra-valued representation.

The pseudocharacter identity is given in (1.1.12.2). The fact that the trace function onthe multiplicative monoid of a matrix algebra satisfies the identity (1.1.12.2) is originallydue to Frobenius [Fro96, §3, 21]. For a modern source, see e.g. [Tay91, Theorem 1(1)].Alternatively, taking Chenevier’s approach, the identity may be deduced as a particularcase of Amitsur’s formula: simply let the homogenous degree i in the homogenous form(1.1.9.8) of Amitsur’s formula be 1. We immediately observe that this is identical to thepseudocharacter condition (1.1.12.2).

To prove (4), we may replace R by R⊗A A[t1, . . . , td] and recall Definition 1.1.8.4 to seethat it will suffice to show that Λi(χ(r)r′) = 0 for all r, r′ ∈ R, 1 ≤ i ≤ d. Applying this toR′ := ZX ⊗Z EX(d), we see that ρuniv χ(r) = 0 in Md(FX(d)) for all r ∈ R′, since χ(r)is the substitution of r into its own characteristic polynomial, which vanishes in Md(FX(d))by the Cayley-Hamilton theorem. Now as Λi factors through ρuniv, we have the result.

Remark 1.1.9.12 (cf. [Che11, Remark 1.13]). Proposition 1.1.9.11(2) (Amitsur’s for-mula) may be proved for homogenous multiplicative polynomial laws into arbitrary associa-tive A-algebras S in the place of commutative A-algebras B. That is, these identities inthe case of determinants of representations are due to Amitsur [Ami80], but they are areparticular instances of facts known to hold in more generality! In particular, an arbitraryhomogenous multiplicative polynomial law is determined by its “characteristic polynomialcoefficients.” These identities are established in this generality by Chenevier in [Che11,Lemma 1.12] (following [RS87]). Here, we have confined our proof to the case that B iscommutative. We refer to Chenevier for the general case.

Remark 1.1.9.13. In contrast to the previous remark, the Cayley-Hamilton identity isspecial not merely to the case that the target of a multiplicative polynomial law is commu-tative, but actually only makes sense in the case of pseudorepresentations (i.e. B = A).

Remark 1.1.9.14. In Proposition 1.1.2.16(3), we showed that certain functions P [α] :R → S characterize a polynomial law P ∈ PdA(R, S). It is quite convenient that when apolynomial law is multiplicative, it can be characterized by what is apparently less data: thed characteristic polynomial coefficient functions Λi,A : R→ S on R alone. Amitsur’s formulauses multiplicativity to express P [α] in terms of Λi,A.

Now we use Amitsur’s formula to show that a pseudorepresentation D : R → A ischaracterized by its characteristic polynomial functions on R, i.e. the function Λi,A : R→ Acontained in the polynomial law Λi : R→ A. In fact, these notions (characteristic polynomialcoefficient polynomial laws Λi, and Amitsur’s formula) make sense even when D : R → Sis a homogenous multiplicative A-polynomial law into a non-commutative A-algebra S (seeRemark 1.1.9.12 below), and we prove this fact in this generality.

Corollary 1.1.9.15 ([Che11, Corollary 1.14]). Let A be a commutative ring, and letR and S be possibly non-commutative A-algebras. Let D : R → S ∈ Md

A(R, S) be a

31

degree d homogenous multiplicative polynomial law. Then characteristic polynomial functionsΛi = Λi,A : R → S of D characterize D. In particular, characteristic polynomial coefficientfunctions characterize D when D is a pseudorepresentation (i.e. A = S).

Proof. We know from Proposition 1.1.2.16(4) that the multiplicative polynomial lawD is characterized by the function

DA[t1,...,td] : R⊗A A[t1, . . . , td]→ S ⊗A A[t1, . . . , td].

We know from the discussion in Remark 1.1.9.12 regarding Chenevier’s proof of Amitsur’sformula that D satisfies Amitsur’s formula. Now Amitsur’s formula (1.1.9.10) allows us toexpress

DA[t1,...,td](r1t1 + · · ·+ rd), (r1, . . . , rd) ∈ Rd

as a sum of monomials in Λi,A(w) and ti with prescribed coefficients 1 and −1, where w is aword in the letters r1, . . . , rd. Therefore the characteristic polynomial functions Λi,A : R→ Scharacterize D, as desired.

Because of its importance, Corollary 1.1.9.15 has been stated succinctly and solitarilyabove. However, there are other consequences of its proof (e.g. consequences of Amitsur’sformula) which are significant. We list them here.

Corollary 1.1.9.16. Let A,R, S,D, and d be as in the previous corollary. Let C ⊂ S bethe sub-A-algebra of S generated by the coefficients Λi(r) of χ(w, t) for all r ∈ R, 1 ≤ i ≤ d.

(1) Then D factors through a (unique) C-valued degree d multiplicative polynomial lawDΛ : R→ C ⊂ S.

(2) The S-valued d-dimensional pseudorepresentation D⊗AB : R⊗AB → B induced byD is induced by the C-valued d-dimensional pseudorepresentation DΛ : R⊗AC → Cinduced by DΛ.

(3) If R is generated over A by some monoid Γ, i.e. R = AΓ, and Λi,A(γ) lie in asub-A-algebra C ⊂ B for all γ ∈ Γ, 1 ≤ i ≤ d, then the conclusion of part (1) holds.

Proof. Part (1) follows from the comment in the proof above that the only factors inthe coefficients other than Λi(w) and ti are 1 and −1. Part (2) follows directly from theproof above, along with the equivalence between multiplicative polynomial laws from R toB and pseudorepresentations from R ⊗A B to B that follows from Corollary 1.1.3.10. Part(3) is a special case of part (1).

1.1.10. Work of Vaccarino, Donkin, Zubkov, and Procesi. In this paragraph wedescribe work leading up Vaccarino’s proof of Theorem 1.1.9.3. We also deduce that if R isa finitely generated A-algebra, then PsRd

R is finite type as an affine A-scheme.The fundamental idea behind the proof of Theorem 1.1.9.3 is the generalization of the

ring of symmetric functions Λ, where Λ is to the singleton set as generalizations of Λ are toother sets. This idea goes at least back to [Don93].

First we review the theory of Λ, corresponding to a singleton set X. Then TSdZ(ZX) ∼=Z[Σ1, . . . ,Σd] =: Λd is the ring of symmetric polynomials on d coefficients (cf. Example1.1.7.7). The ring of symmetric functions is Λ := lim←−d Λd, where the maps are given by

ld : Λd → Λd−1, (x1, . . . , xd) 7→ (x1, . . . , xd−1).

One key fact about this limit presentation is its behavior under the filtration by homogenouspolynomial degree, which we will denote by n here. Let Λn

d denote grnΛd for n ≥ 0. Then

32

for d ≥ n, the composition of the li for n+ 1 ≤ i ≤ d induces an isomorphism Λnd∼→ Λn

n. Forexample, the “trace” Σ1 = x1 + · · ·+ xd ∈ Λ1

d is a generator for Λ1d for all d ≥ 1.

Now we generalize this construction of Λ, starting with a finite set X. Write X+ forthe associated monoid of words with letters in X. As Vaccarino proves these results for anarbitrary commutative base ring A, we will replace Z above with A. We can assign eachelement of X degree 1, which induces a degree, which we will index by n ≥ 0, on TSdA(AX)for any d. Write the nth graded piece as TSdA(AX)n. With the analogous maps of graded(non-commutative) A-algebras ld : TSdA(AX)→ TSd−1

A (AX), the inverse limit

TSA(AX) := lim←−d

TSdA(AX)

stabilizes on each graded piece, so that (cf. [Vac08, Corollary 5.5])

TSA(AX)n := lim←−d

TSdA(AX)n ∼= TSnA(AX)n.

Therefore TSA(AX) is a graded A-algebra with each homogenous summand being finitelygenerated as an A-module. Moreover, all of these objects are free A-modules with an explicitbasis that we do not require here [Vac08, Propostion 3.12]. It will be useful to have a setof generators of TSdA(AX) as a A-algebra, however. Recall the notation of the proof ofProposition 1.1.4.5, in particular the basis element eK for TSdA(AX), where K = K(w, i)is the equivalence class of tensors including

eK(w,i) := w⊗i ⊗ 1⊗

d−i ∈ TSdA(AX)as its special representative for some w ∈ X+, i ≤ d. For future reference, it will be helpful

to record that if e(d)K(w,i) ∈ TSdA(AX), where we make the degree d of the basis element

explicit, then ld : TSdA(AX)→ TSd−1A (AX) is given by the formula

(1.1.10.1) ld : e(d)K(w,i) 7→

e

(d−1)K(w,i) if i < d,

0 if i = d,

which is directly analogous to the maps in the theory of ring of symmetric functions. As anA-algebra, TSdA(AX) is generated by eK(w,i) as i varies over positive integers less than d andw varies over elements of X+ that are “primitive,” i.e. not proper powers of another word[Vac08, Theorem 4.10]. By the stabilization of the grading discussed above, these eK(w,i)

have a canonical preimage in TSA(AX), and these preimages generate the A-algebra asf primitive and i ranges over all positive integers. We summarize our knowledge in thisproposition

Proposition 1.1.10.2. The graded A-algebra TSA(AX) is free as an A-module andgenerated as an A-algebra by eK(w,i) as w ranges over primitive words in X+ and i rangesover positive integers.

All of these statements hold true after replacing each of these A-algebras with theirabelianizations,3 and although this is non-trivial, in fact even more is true: TSA(AX)ab

is a polynomial ring over A! We record this result in Theorem 1.1.10.8 below, but first we

3Indeed, it is the freeness of ΓdZ(ZX)ab as a Z-module that is the fundamental input from the work ofVaccarino et. al. that Chenevier needs to establish the Cayley-Hamilton identity. But this comes part-and-parcel with the rest of these results, cf. [Che11, Remark 1.16].

33

explain the proof, as we will accomplish our main task of proving Theorem 1.1.9.3 along theway.

Recall the generic matrices representations ρunivd : AX → Md(FX(d)A) for each d ≥ 1,

where FX(d)A denotes FX(d)⊗Z A. The determinant of ρunivd is a d-dimensional pseudorep-

resentation of AX, inducing a canonical ring homomorphism

(1.1.10.3) δd : TSdA(AX) −→ FX(d)A,

using TSdA(AX) in place of ΓdA(AX) in light of Proposition 1.1.4.5. He then observes in[Vac08, Proposition 5.19] that

(1.1.10.4) δd(eK(w,i)) = Λi(ρuniv(w))

for all primitive w and 1 ≤ i ≤ d, where Λi : Md(FX(d)A) → EX(d)A is the ith coefficientof the standard characteristic polynomial on the matrix algebra and, recall, EX(d)A is thesub-A-algebra of FX(d)A generated by coefficients of characteristic polynomials of the imageof ρuniv

d . As the eK(w,i) generate TSdA(AX), this shows that the characteristic polynomial

coefficient functions generate the image of TSdA(AX), i.e. the image of δd is preciselyEX(d)A. Since the image of δd is a commutative algebra, we have a surjective induced map

(1.1.10.5) δabd : TSdA(AX)ab EX(d)A.

Remark 1.1.10.6. The line of argument that we have just concluded is sufficient to proveCorollary 1.1.9.15.

Following [Vac08, §5.1.3], we extend this representation and the maps δd to the limitas d → ∞. First we filter FX(d)A and EX(d)A by degree denoted n, where the generatorsxij for x ∈ X, 1 ≤ i, j ≤ d are given degree 1. With the notation of Definition 1.1.9.1, letωd : FX(d)A FX(d− 1)A via

xij 7→xij if i, j < d0 if i = d or j = d.

This induces a map (ωd)d : Md(FX(d)A)→Md(FX(d− 1)A) such that

(ωd)d ρunivd =

(ρunivd−1 0d−1×1

01×d−1 0

).

We observe that Λ(d)i (ωd)d ρuniv

d = Λ(d−1)i ρuniv

d−1 for d ≥ 1, where the superscript on thecharacteristic polynomial coefficient function indicates the dimension of the matrix algebra

on which it is defined. As a result, since the image of Λ(d)i ρuniv

d generates EX(d)A, we have awell-defined induced map ωd : EX(d)A → EX(d−1A) on the A-subalgebra EX(d)A ⊂ FX(d)A.Therefore the maps ωd induce limits of graded A-algebras

FX,A := lim←−d

FX(d)A ⊃ EX,A := lim←−d

EX(d)A

with the same stabilization properties for the filtration by degree as discussed above forthe limit defining TSA(AX). In particular, for any w ∈ X+, there is a well definedcharacteristic polynomial coefficient Λi(ρ

univ(w)) ∈ EX,A, where Λi(ρuniv(w)) has bounded

degree i · `(w) where `(w) is the length of w. Strictly speaking, Λi ρuniv := lim←−d Λ(d)i ρuniv

d .

34

Define δ : TSA(AX) EX,A by

· · · // TSdA(AX) ld //

δd

TSd−1A (AX)

ld−1 //

δd−1

· · ·

· · · // EX(d)Aωd // EX(d− 1)A

ωd−1 // · · ·

where the fact that δ is a surjection must be deduced from the fact that each δd is a surjectionby definition, along with a study of the gradings ([Vac08, Lemma 5.22]). The generating seteK(w,i) for TSA(AX) of Proposition 1.1.10.2 and the calculation of (1.1.10.4) shows thatthe characteristic polynomial coefficients Λi(ρ

univ(w)) generate EX,A, where i varies overpositive integers and w ∈ X+ vary over primitive words. Of course, δ factors through

TSA(AX) −→ TSA(AX)ab,

and our goal is to show that

δab : TSA(AX)ab −→ EX,A

is an isomorphism. This will follow from this result of Donkin:

Theorem 1.1.10.7 ([Don93, §3(10)]). The ring EX,A is a polynomial ring over A with freegenerators Λi(ρ

univ(w)), where w varies over a set Ψ representatives of equivalence classesof primitive words, where the equivalence relation is cyclic permutation.

Now we can prove that δab is an isomorphism.

Theorem 1.1.10.8 ([Vac08, Theorem 5.23]). The map of graded A-algebras

δab : TSA(AX)ab −→ EX,A

is an isomorphism, and, consequently, the commutative A-algebra TSA(AX)ab is a poly-nomial ring over A with generators eK(f,i) where i ≥ 1 and f varies over Ψ.

Proof. We know that δab is a surjection. As EX,A is a free polynomial A-algebra, thereexists a section s : EX,A → TSA(AX)ab sending Λi(ρ

univ(w)) to the image of eK(w,i) in theabelianization, where i ≥ 1 and w varies over the representatives of the equivalence classesmentioned in Theorem 1.1.10.7. By [Vac08, Corollary 5.12], eKw,i, w ∈ Ψ are sufficient togenerate TSA(AX)ab. Therefore s is surjective, and δab is an isomorphism.

Now, our goal is to deduce from Theorem 1.1.10.8 that δd is an isomorphism as well.Here, Vaccarino’s remaining work is to apply work of Procesi, Razmyslov, and Zubkov,whose background we now explain.

The issue we must confront is the determination ideal of relations that the free generators

Λi(w) := Λi(ρuniv(w)) of EX,A satisfy when they are projected to Λ

(d)i (w) ∈ EX(d)A. Clearly

if i > d, then Λi(w) ≡ 0 ∈ EX(d)A, and the Λ(d)i (w) generate EX(d)A. But are there

further relations? And are there more relations among Λ(d)i (w) ∈ EX(d)A than among

eK(w,i) ∈ TSdA(AX)?When A is an algebraically closed field of characteristic zero, this question was answered

by Procesi [Pro76, Theorem 4.6(a)] and Razmyslov [Raz74]: the kernel of EX,A EX(d)Ais generated as an ideal by Λd+1(w) as w varies over representatives of equivalence classesof primitive words. For an arbitrary infinite field A, it was shown by Zubkov [Zub96, Main

35

Theorem] that the kernel of EX,A EX(d)A is the ideal generated by

(1.1.10.9) Λi(w) | i > d, w primitive.The answers to the analogous questions for TSA(AX) TSdA(AX) are easier, and

A may be an arbitrary commutative ring: we know that TSA(AX) is a graded polynomialalgebra in the variables eK(w,i) | w ∈ Ψ, i ≥ 1. By examining the explicit presentation ofthe maps ld composing the limit defining AX in (1.1.10.1), we see that

0 −→ (eK(w,i) : w primitive, i > d) −→ TSA(AX) −→ TSdA(AX) −→ 0

is exact. Applying Lemma 1.1.7.12, the sequence

(1.1.10.10) 0→ abTS(eK(w,i) : w primitive, i > d)→ TSA(AX)ab → TSdA(AX)ab → 0

is still exact.Therefore, when A is an infinite field, using Zubkov’s result in (1.1.10.9) along with

(1.1.10.10) and the isomorphism of Theorem 1.1.10.8, we have that

(1.1.10.11)

TSdA(AX)ab ∼= A[eK(w,i) : i ≥ 1, w ∈ Ψ]/abTS(eK(w,i) : i > d, w primitive∼= EX,A/(Λi(w) | i > d, w primitive)∼= EX(d)A.

Vaccarino’s final task is to show that this isomorphism over infinite fields A implies thatthe isomorphism holds in the case A = Z. To explain this last step, we introduce some morebackground on the interest in these objects, culminating in a result over Z that we will needto finish the proof of Theorem 1.1.9.3.

Recall the universal representation

ρuniv = ρunivd : AX −→Md(FX(d)A)

from Definition 1.1.9.1. The adjoint action of PGLd(A) on Md(A) for all commutative ringsA induces an action of the group scheme PGLd/ SpecZ on SpecFX(d) = SpecFX(d)Z, withg ∈ PGLd(A) sending xij for x ∈ X, 1 ≤ i, j ≤ d to the element of FX(d)A appearing

in the (i, j)-coordinate after conjugation by g. Clearly EX(d)A ⊂ FX(d)PGLd(A)A for all A,

because characteristic polynomial coefficients are invariant under conjugation. Is this mapan isomorphism?

This question was first investigated for A an algebraically closed field of characteristiczero, and then for positive characteristic algebraically closed fields and A = Z. The motivat-ing question was to describe the invariant theory of n-tuples of d× d-matrices (m1, . . . ,mn).That is, what is the subring of regular functions on the affine variety Mn

d = Md × · · · ×Md

invariant under the diagonal action of PGLd by conjugation on Md × · · · ×Md? M. Artinconjectured4 that the subring of conjugation-invariant regular functions were generated bytraces of products of these n matrices, i.e. for some finite word w in the alphabet 1, . . . , nwith letters wi, the regular function

Tr(mw1 ·mw2 · · · · ·mwn)

on Mnd . In positive characteristic, one conjectures that such functions will generate the

invariant subring once other characteristic polynomial coefficients Λi, 1 ≤ i ≤ d are alsoallowed. In other words, the conjecture is that EX(d) = FX(d)PGLd . This conjecture canbe extended over arbitrary bases. To be clear, over the base ring A, FX(d)PGLd

A denotes the

4For the attribution of this conjecture, see [Pro76, Introduction].

36

co-invariants of the co-action of the coordinate ring of PGLd/A

FX(d)A −→ FX(d)A ⊗A A[PGLd],

i.e. those f ∈ FX(d) such that its image is f ⊗ 1. Also, set FX(d)PGLd := FX(d)PGLdZ .

This is the main result of Donkin [Don92, §3] over arbitrary algebraically closed fieldsand over Z (depending on his integrality result [Don93]); this was also proved by Zubkov[Zub94]. This followed a proof by Procesi [Pro67] and, independently, Sibirski [Sib67], ofArtin’s conjecture in the characteristic zero case. Here is the key result of Donkin’s work forour purposes.

Theorem 1.1.10.12 ([Don92, §3.1]). For A = Z and d ≥ 1, the map EX(d) −→FX(d)PGLd is an isomorphism, and, for every algebraically closed field k, induces an iso-morphism

EX(d)⊗Z k∼−→ FX(d)PGLd

k.

Vaccarino uses this theorem along with the following argument (cf. [Vac08, Theorem 6.1])to complete the proof of Theorem 1.1.9.3.

Proof. (Theorem 1.1.9.3) Let A be a commutative ring. By Corollary 1.1.3.10(1),

we have an isomorphism TSdA(AX) ∼→ TSdZ(ZX) ⊗Z A. By the universal property ofabelianization, the map

abTSZ ⊗ 1A : TSdZ(ZX)⊗Z A −→ TSdZ(ZX)ab ⊗Z A

can be factored through the abelianization TSdA(AX)→ TSdA(AX)ab, making the com-mutative diagram

(1.1.10.13) TSdA(AX) ∼ //

abTSA

TSdZ(ZX)⊗Z A

abTSZ⊗1A

TSdA(AX)ab // TSdZ(ZX)ab ⊗Z A

where the bottom horizontal arrow is surjective. Letting A = k, an algebraically closed field,this bottom horizontal arrow is the top arrow in the commutative diagram

TSdk(kX)ab

∼=

// TSdZ(ZX)ab ⊗Z k

δabd ⊗1k

(FX(d)k)PGLd

∼= // EX(d)⊗Z k

where the composite map from the top left to the bottom right is known to be an isomorphismby (1.1.10.11) and the bottom horizontal arrow is known to be an isomorphism by Theorem1.1.10.12. Since we know from (1.1.10.13) that the top horizontal arrow is surjective, andthe right vertical arrow is surjective since it is obtained by ⊗Zk from the surjective map δab

d

of (1.1.10.5), all of the maps in the diagram are isomorphisms.Therefore we have a surjective map of graded rings

(1.1.10.14) TSdZ(ZX)ab −→ EX(d)

37

that becomes an isomorphism after tensoring by any algebraically closed field. Each gradedcomponent TSdZ(ZX)ab

n , EX(d)n of each ring is a finite Z-module, and these finite Z-modules are free because they are submodules of the polynomial algebra EX , and there-fore torsion-free. As these finite free Z-modules become isomorphic after tensoring by anyalgebraically closed field, they must be of the same rank and therefore (1.1.10.14) is anisomorphism.

Now we discuss the finite generation of ΓdA(R)ab over A. The invariant theoretic contentabove will be very useful for this. We can now show the that ΓdZ(ZX)ab is finitely generatedover Z when X is finite, from which we can deduce that PsRd

R is finitely type as an affineA-scheme when R is a finitely generated A-algebra. We follow Chenevier, using the invarianttheoretic content above with the input of geometric invariant theory.

Theorem 1.1.10.15 ([Che11, Proposition 2.38]). Let A be a commutative Noetherianring, let R be a finitely generated A-algebra, and let d ≥ 0. Then ΓdA(R)ab is finitely generatedas an A-algebra.

Proof. Let X be a finite set an let m = |X|. As R is finitely generated over A, thereexists a surjective A-algebra homomorphism

AX R,

and therefore also a surjective A-algebra homomorphism ΓdA(AX)ab ΓdA(R)ab, where thesurjectivity follows from Corollary 1.1.3.14 and Lemma 1.1.7.12. Therefore we are reducedto the case that R = AX. As ΓdA(AX)ab ∼= ΓdZ(ZX)ab ⊗Z A by Corollary 1.1.3.10, wefurther reduce to the case A = Z.

Our main achievement of this section, Theorem 1.1.9.3, shows that the determinant ofρuniv is a pseudorepresentation inducing an isomorphism ΓdZ(ZX)ab ∼→ EX(d). By Theorem1.1.10.12, EX(d) ∼= FX(d)PGLd . By the main theorems of geometric invariant theory (see forexample [Alp10, Main Theorem, (4)] or the original source [Ses77, Theorem 2]), the factthat FX(d) is finitely generated over Z implies that FX(d)PGLd is finitely generated over Zas well.

Remark 1.1.10.16. We will often assume the assumptions of Theorem 1.1.10.15, so thatPsRd

R is an affine Noetherian A-scheme. Later, we will see that these assumptions are alsonecessary in order to know moduli spaces of representations and are finite type over SpecA.

1.1.11. A Direct Sum Operation on Pseudorepresentations. Given two repre-sentations of an A-algebra R, one can form a representation out of their direct sum. In thisparagraph, we study the analogy of this construction for pseudorepresentations, and verifythat this operation behaves well with respect to dimension.

Let R1, R2 be A-algebras, and let B be a commutative A-algebra. We know from Corol-lary 1.1.3.10(3) along with Theorem 1.1.6.5 that we have an isomorphism of A-algebras

(1.1.11.1) ΓdA(R1 ×R2)ab ∼−→∏

d1+d2=d

Γd1A (R1)ab ⊗A Γd2

A (R2)ab.

By representability, this corresponds to a binary operation, associating two multiplicativeA-polynomial laws

D1 : R1 −→ A, D2 : R2 −→ A,

38

which are homogenous of degree di respectively, to their product, which is a multiplicativeA-polynomial law

D1 ·D2 : R1 ×R2 −→ A

of degree d = d1 + d2. One can check that the construction is

(1.1.11.2)D1 ⊕D2 : R1 ×R2 −→ A

(r1, r2) 7→ D1(r1) ·D2(r2),

which is compatible with ⊗AB, and thereby a polynomial law. One can also quickly see thatthis polynomial law is multiplicative of degree d = d1 + d2 (cf. 1.1.11.5).

Remark 1.1.11.3. It is important to notice that the case of degree 0 homogenous mul-tiplicative polynomial laws play an important role in the isomorphisms above: for example,in (1.1.11.1), the d1, d2 must vary over all non-negative integers such that d1 + d2 = d. Wealso see the importance of zero-dimensional pseudorepresentations having constant value 1.

It is natural, since pseudorepresentations are sometimes constructed by taking determi-nants, to think of this operation as a product. However, we will call it a sum, either byanalogy to the data of the trace function that a pseudorepresentation holds, or by observingthat if we have two representations

R1 −→Md1(A), R2 −→Md2(A),

then there is a direct sum representation

R1 ×R2 −→Md1(A)×Md2(A) →Md1+d2(A)

which is compatible with the construction above by taking the pseudorepresentations inducedby the determinants of the three representations.

Remark 1.1.11.4. We also choose to call this operation a sum ⊕ on pseudorepresenta-tions because some preliminary calculations suggest that if R has the structure of a (cocom-mutative) Hopf algebra, there is a (commutative) tensor product operation ⊗ on pseudorep-resentations which decategorifies the tensor product of representations of R.

We summarize our discussion about the sum in this proposition, also adding the basicfact that the degree of a homogenous polynomial law into a commutative ring is locallyconstant; then we know that we are not making any restriction by studying homogenousmultiplicative polynomial laws of a given degree.

Proposition 1.1.11.5 (Following [Che11, Lemma 2.2]). With R1, R2 being A-algebras,let B be a commutative A-algebra and let Di : Ri → B be a multiplicative A-polynomial laws.If SpecB is connected, then

(1) D1 (resp. D2) is homogenous of some degree d ≥ 0.(2) any degree d homogenous multiplicative A-polynomial law D : R1 × R2 → B is the

sum, D1⊕D2, of two unique multiplicative homogenous polynomial laws Di : Ri → Aof degree di, with d1 + d2 = d.

Part (2) is also proved in [Che11, Lemma 2.2(iii)].

Proof. Part (1) is precisely Theorem 1.1.7.4(2).

39

To prove (2), simply observe that as SpecB is connected, its image in PsRdR1×R2

must beconfined to one of the elements of the disjoint union

PsRdR1×R2

∼−→∐

d1+d2=d

PsRd1R1×SpecA PsRd2

R2

induced by (1.1.11.1).

Now we set R1 = R2 = R, so that the work above amounts to the analogue in thecategory of pseudorepresentations of the construction of the R × R-module M ⊕ N out oftwo R-modules M,N , where the first copy of R acts on N trivially and the second copy of Racts on M trivially. To construct from this R×R module the direct sum R-module M ⊕N ,we simply compose with the diagonal embedding

R∆−→ R×R.

This construction inspires the construction of the direct sum of pseudorepresentations.

Definition 1.1.11.6. Let R be an A-algebra, and let D1, D2 be pseudorepresentationsof dimension d1, d2 of R over A. Set d = d1 + d2. Then the direct sum pseudorepresentationD := D1 ⊕D2 of R over A is given by the d-dimensional homogenous polynomial law suchthat for each commutative A-algebra B,

DB(x) = D1,B(x) ·D2,B(x) ∀x ∈ R⊗A B.

We take note of the basic properties of this operation.

Lemma 1.1.11.7. Let R be an A-algebra, and let d1, d2, and d be non-negative integerssuch that d1 + d2 = d. Then

(1) The operation⊕ : PsRd1

R ×SpecA PsRd2R −→ PsRd

R

is a morphism in the category of affine A-schemes, corresponding to the homomor-phism of commutative A-algebras

ΓdA(R)ab Γd(∆)−→ ΓdA(R×R)ab(1.1.11.1) Γd1

A (R)ab ⊗A Γd2A (R2)ab.

(2) If D1 ∈ PsRd1R (B) and PsRd2

R (B) are induced from d-dimensional B-valued repre-sentations of R, ρ1 of dimension d1 and ρ2 of dimension d2 respectively, then thedet (ρ1 ⊕ ρ2) ∼= D1 ⊕D2. In other words, the direct sum operations on representa-tions and pseudorepresentations commute with the map det from representations topseudorepresentations.

Proof. For (1), simply compose (1.1.11.2) and ∆, and note that this is the same as thedirect sum given in Definition 1.1.11.6.

To prove (2), we note that the determinant of a direct sum of representations is equal tothe product of the determinants of the representations.

The structures above induce a commutative monoid structure on the functor of all pseu-dorepresentations.

Definition 1.1.11.8. Let R be an A-algebra. Then write PsR+R for the SpecA-scheme

in commutative monoidsPsR+

R :=∐d≥0

PsRdR,

40

where the group operation is

⊕ : PsR+R ×SpecA PsR+

R −→ PsR+R

and the identity section isSpecA ∼= PsR0

R → PsR+R.

Later in Theorem 1.3.1.1, we will see that when A ∼= k is an algebraically closed field,the commutative monoid PsR+

R(k) will be the Grothendieck semigroup of the category ofrepresentations of R.

1.1.12. Relation to Pseudocharacters. In this paragraph, we describe a previousversion of a pseudorepresentation, which is also commonly known as a pseudorepresenta-tion. This is a pseudocharacter, which is a function on an algebra or multiplicative monoidsatisfying the identities one expects of the trace function of a matrix algebra.

Definition 1.1.12.1 (cf. [Tay91, §1.1], [Nys96, Rou96]). Let Γ be a monoid and let Abe a commutative ring. Let R be an A-algebra. A pseudocharacter of Γ over A of dimensiond is the data of a function T : Γ→ A such that

(1) T (1) = d,(2) T is central, i.e. T (γ1γ2) = T (γ2γ1) for all γ1, γ2 ∈ Γ, and(3) the d-dimensional pseudocharacter identity holds:

(1.1.12.2)∑

σ∈Sd+1

sgn(σ)Tσ(γ1, . . . , γd+1) for all γ1, . . . , γd+1 ∈ Γ,

where Sd+1 is the symmetric group on d+ 1 letters and Tσ is the function given by

Tσ : Γd+1 −→ A

(γ1, . . . , γd+1) 7→s∏j=1

T (γi(j)1· · · γ

i(j)rj

),

where σ has cycle decomposition

σ = (i(1)1 . . . i(1)

r1)(i

(2)1 . . . i(2)

r2) . . . (i

(s)1 . . . i(s)rs ).

The definition of a pseudocharacter for R is identical, using the multiplicative monoid of R,except that we impose the additional condition that T be A-linear.

Taylor [Tay91] gave the definition of pseudorepresentation above, following on Wiles’definition for two-dimensional representations [Wil88]. That the identity (1.1.12.2) is satis-fied by a trace function of a representation is due to Frobenius [Fro96], and Procesi [Pro76,Theorem 1.2] showed that this is the only identity that a central function needs to satisfyin order to correspond to an invariant (by the adjoint action) function on a space of repre-sentations. Taylor used this result to show that pseudorepresentations over an algebraicallyclosed field of characteristic zero are in natural bijection with semisimple characteristic zerorepresentations up to isomorphism. Rouquier [Rou96] extended this to the case that thecharacteristic of the field is either 0 or greater than the dimension of the pseudocharacter.We will give Chenevier’s [Che11] extension of this theorem to arbitrary characteristic, whichis achieved by replacing pseudocharacters with pseudorepresentations, in Theorem 1.3.1.1.

Carayol [Car94] showed that the deformations of an absolutely irreducible representationover a field are determined by the induced deformation of its pseudocharacter, where this

41

deformation is given by the trace function of the representation. Nyssen [Nys96, Theorem1] and Rouquier [Rou96, Theorem 5.1] proved a converse, showing that deformations of apseudocharacter to a henselian local ring determine a unique (up to isomorphism) defor-mation of its associated semisimple representation. Definition 1.3.4.1 describes absolutelyirreducible pseudorepresentations, and we give Chenevier’s an analogous result for pseu-dorepresentations to the result of Cayayol, Nyssen, and Rouquier’s work in the in Theorem2.1.3.3.

To what extent are pseudocharacters and pseudorepresentations comparable? This propo-sition, due to Chenevier, gives the state of knowledge on this question.

Proposition 1.1.12.3 ([Che11, Propositions 1.27 and 1.29]). Let A be a commutativering and let R be an A-algebra. To each d-dimensional pseudorepresentation D : R→ A, weassociate to D its trace function function T = Λ1,A : R→ A via Definition 1.1.8.2.

(1) T is a pseudocharacter of dimension d; in particular, it satisfies (1.1.12.2).(2) The association of determinants to pseudocharacters is injective.(3) If (2d)! ∈ A×, then the association is bijective.

Proof. The first part is precisely Proposition 1.1.9.11(3). See [Che11, Propositions 1.27and 1.29] for parts (2) and (3).

We will use the theory of pseudocharacters in §2.3 in order to apply Bellaıche-Chenevier’sdefinition of generalized matrix algebra. We propose a notion of generalized matrix algebrawith respect to pseudorepresentations instead of pseudocharacters in Remark 2.3.3.6. How-ever, when we do this, we will restrict ourselves to the case that (2d)! is invertible in ourcoefficient rings, so that we can join our theory of pseudorepresentations with the theory ofgeneralized matrix algebras. Proposition 1.1.12.3(3) shows that this is sensible.

1.2. Cayley-Hamilton Pseudorepresentations

Recall from Definition 1.1.8.5 that a pseudorepresentation D : R −→ A is called Cayley-Hamilton if the homogenous degree d pseudorepresentation

χ = χD : r 7→ rd − Λ1(r)rd−1 + Λ2(r)rd−2 + · · ·+ (−1)dΛd(r)

vanishes identically, i.e. every element of R satisfies its own characteristic polynomial, justas if R were a matrix algebra. We also say that (R,D) is a Cayley-Hamilton A-algebra.Cayley-Hamilton algebras have several special properties which we will explore here. We aremotivated by exploring to what extent R has similarities to matrix algebras. For example,Procesi [Pro87] proved that in characteristic zero, a Cayley-Hamilton A-algebra admits anembedding into a matrix algebra Md(B) for some commutative A-algebra B.

While the material of this section is mostly due to Chenevier [Che11], our main newcontribution is the application of polynomial invariant ring (PI ring) theory to show thatCayley-Hamilton algebras are finite over their pseudorepresentation algebra OPsRdR

, and inparticular finite over their center. This allows us to strengthen one of Chenevier’s results.

1.2.1. Properties of Cayley-Hamilton Algebras. We freely use the notation of Def-inition 1.1.8.5. One of the most basic properties of the two-sided ideal CH(D) ⊂ R is thefollowing lemma, showing that any pseudorepresentation factors through a Cayley-Hamiltonalgebra and that faithful pseudorepresentations are Cayley-Hamilton.

42

Lemma 1.2.1.1 ([Che11, Lemma 1.21]). With A, R, D, and d as usual, ker(D) containsCH(D). In particular, if D is faithful, then (R,D) is Cayley-Hamilton.

Proof. We have proved the “Cayley-Hamilton identity for pseudorepresentations” inProposition 1.1.9.11(4), namely

D(1 + χ[α](r1, . . . , rd) · r) = 1

for any α ∈ Idd , (r1, . . . , rd) ∈ Rd, and r ∈ R. It remains to show that this holds true afterreplacing r with an element of R ⊗A B for B any commutative A-algebra. This follows

from the fact that, for a given α, the functions χ[α]B : (R ⊗A B)d → R associated to the

pseudorepresentation D ⊗A B : R⊗A B → B belong to the commutative diagram

Rd χ[α]

//

R

(R⊗A B)d

χ[α]B // R⊗A B

Example 1.2.1.2 ([Che11, Example 1.20]). Consider a matrix algebra Md(A) over acommutative ring A, with its standard d-dimensional pseudorepresentation det coming fromthe determinant Md(A) → A. Of course, this pseudorepresentation is Cayley-Hamilton, asevery matrix satisfies its characteristic polynomial by the Cayley-Hamilton theorem. It isalso faithful, since for any 0 6= r ∈ Md(A) there exists r′ ∈ Md(A) such that the char-acteristic polynomial of rr′ is not td. Consider now the restriction D : Td(A) → A ofdet to the A-subalgebra Td(A) ⊂ Md(A) of upper triangular matrices. We see that D isstill Cayley-Hamilton, illustrating the general fact that the restriction of a Cayley-Hamiltonpseudorepresentation to a subalgebra remains Cayley-Hamilton. However, this example alsoillustrates that the “faithful” property of a pseudorepresentation is not stable under restric-tion to a subalgebra. For det is faithful, but the kernel of D is precisely the two-sided idealof strictly upper triangular matrices in Td(A).

We record the following lemma on the decomposition of a pseudorepresentation by idem-potents. Recall that an idempotent e ∈ R induces a decomposition eRe ⊕ (1 − e)R(1 − e)which is a A-subalgebra of R isomorphic to eRe× (1− e)R(1− e) via the natural map

(1.2.1.3) x 7→ (ex, (1− e)x).

Also recall that a set of idempotents is called orthogonal provided that the product of anypair of distinct elements of the set is zero. Note that not all of this lemma depends on Dbeing Cayley-Hamilton.

Lemma 1.2.1.4 ([Che11, Lemma 2.4]). Assume that SpecA is connected and let e ∈ Rbe an idempotent element. Let D : R→ A be a d-dimensional pseudorepresentation.

(1) The polynomial law De : eRe → A defined by r 7→ D(r + 1 − e) is a pseudorepre-sentation whose dimension r(e) satisfies r(e) ≤ d.

(2) We have r(e) + r(1 − e) = d, and the restriction of D to the A-subalgebra eRe ⊕(1− e)R(1− e) is the direct sum pseudorepresentation DeD1−e of (1.1.11.2).

(3) If D is Cayley-Hamilton (resp. faithful), then so is De.(4) Assume that D is Cayley-Hamilton. Then e = 1 (resp. e = 0) if and only if D(e) = 1

(resp. r(e) = 0). Let e1, . . . , es be a family of nonzero orthogonal idempotents of R.

43

Then s ≤ d, and we have an inequality∑s

i=1 r(ei) ≤ d, which is an equality if andonly if e1 + e2 + · · ·+ es = 1.

Proof. Write S1 = eRe and S2 = (1 − e)R(1 − e). Let S be the A-subalgebra S =S1 ⊕ S2 ⊂ R. As noted above, (1.2.1.3) induces an isomorphism with S1 × S2. Now parts(1) and (2) follow directly from Proposition 1.1.11.5.

Assume that D is faithful. Note that for any commutative A-algebra B, the B-algebraeRe ⊗A B is naturally isomorphic to a direct summand e(R ⊗A B)e of R ⊗A B. Chooser ∈ ker(De) ⊂ eRe ⊂ R. Using the characterization of the kernel in Lemma 1.1.6.6, we havefor any r′ ∈ R⊗A B that

D(1 + rr′) = D(1 + erer′) = D(1− e+ e+ erer′) = De(e+ erer′) = 1.

Therefore r ∈ ker(D) so r = 0 by assumption.Assume that D is Cayley-Hamilton. For r ∈ R ⊗A B, we have the Cayley-Hamilton

identity χD(r, r) = 0. From part (2), we know that

χD(r, t) = χDe(er, t)χD1−e((1− e)r, t) ∈ B[t].

For r ∈ e(R⊗A B)e, we apply the Cayley-Hamilton identity for χD to r (resp. r + 1− e) tofind that

χDe(er, r)rd2 = 0, resp. χDe(e(r + 1− e), r + 1− e)(r − 1)d2 = 0.

As the ideal of B[t] generated by td2 and (t− 1)d2 is B[t], we get De(r, r) = 0, showing thatDe is Cayley-Hamilton.

Let us show part (4). It is always the case that χD(e, e)−D(e) ∈ Ae ⊂ R. If D is Cayley-Hamilton and D(e) = 1, then e is a unit in A (see (1.2.3.3) for this fact) and therefore e = 1.If r(e) = 0, then De(·) = D(·+ 1− e) is a determinant of degree 0 on eRe, and is thereforeconstant and equal to 1. In particular, D(1 − e) = 1 so e = 0 by the argument above. Forthe last claim of part (4), set es+1 = 1 − (e1 + · · · + es). Note that 1 ≤ r(ei) ≤ d for eachei, since ei 6= 0 and therefore r(ei) 6= 0 for each i. However,

∑s+11 r(ei) = d by applying part

(2) s times. This proves the last claim in (4).

Lemma 1.2.1.5 ([Che11, Lemma 2.6]). Let D : R → A be a 1-dimensional Cayley-Hamilton pseudorepresentation. Then R = A and D is the identity map.

Proof. For each r ∈ R, χ(r, t) = t −D(r). As r satisfies its characteristic polynomialand D is A-linear, the lemma follows.

1.2.2. Background in PI Ring Theory. Our main aim in this paragraph is to applythe theory of polynomial identity rings to prove that a Cayley-Hamilton A-algebra (R,D)is often finite as a module over A. One implication of this is that all representations of anarbitrary finitely generated A-algebra R of a fixed dimension d simultaneously factor throughan algebra that is finite over its center.

We begin with a short review of the theory of polynomial invariant algebras over acommutative ring A, following Procesi’s book [Pro73]. We will use the notation Axs todenote the free (non-commutative) A-algebra on a set X.

Definition 1.2.2.1. Let R be an A-algebra.

(1) An ideal I ⊂ Axs is called a T -ideal if, for any endomorphism ϕ : Axs → Axs,we have ϕ(I) ⊆ I.

44

(2) The setI = f(xs) ∈ Axs | f(rs) = 0 for all rs ∈ R

is called the T -ideal of polynomial identities of R.(3) A T -ideal I ⊂ Axs is called a proper T -ideal provided that it is not contained in

Jxs for any ideal J 6= A of A.(4) We call R a polynomial ideal algebra or PI-algebra if the T -ideal I of polynomial

identities of R is proper.

Since every element of a degree d Cayley-Hamilton A-algebra satisfies its own degree dcharacteristic polynomial, which is monic and degree d in A[t], it is a PI A-algebra becauseof the following fact.

Proposition 1.2.2.2 ([Pro73, Proposition 3.22]). Let d be a positive integer. Then thereexists a proper polynomial identity such that for any commutative ring A and any A-algebraR, R satisfies this polynomial identity if every element of R is integral over A of degreebounded by d. In particular, such an algebra R is a PI A-algebra.

Proof. We will specify this polynomial identity and leave it to the reader to complete theproof or look up the reference. Let Pn for n ≥ 1 be the polynomial in the (noncommutative)free algebra over Z generated by n indeterminates x1, . . . , xn given by

Pn(x1, . . . , xn) =∑σ∈Sn

sgn(σ)xσ(1)xσ(2) · · ·xσ(n),

where Sn is the symmetric group on n letters and sgn is the signature character sgn : Sn →±1. Define f(x, y) in the (noncommutative) free algebra over Z by

f(x, y) = Pd+1(ydx, yd−1x, yd−2x, . . . , yx, x).

Then f is a proper polynomial identity whose existence is asserted in the statement of theproposition.

When A is Noetherian and R is finitely generated a A-algebra and Cayley-Hamilton, thefollowing fact will allow us to conclude immediately that R is finite as an A-module.

Theorem 1.2.2.3 ([Pro73, Theorem 2.7]). Let R be a finitely generated PI algebra overa commutative Noetherian ring A. Then if R is integral over A, it is also finite as a moduleover A.

However, in some particular cases relevant to our investigation of Cayley-Hamilton al-gebras, we will be able to establish module finiteness of R over A when R satisfies weakerconditions than the conditions of Theorem 1.2.2.3.5 In order to accomplish this, it will beparticularly important to show that if R is a nil algebra (i.e. every element is nilpotent; inparticular, a nil algebra does not have a unit) of bounded nil degree over a field k, then Ris finite dimensional as a k-vector space. The following theorems will be very useful to thisend.

The first important theorem is known as the Nagata-Higman theorem.

5The most important example will be Theorem 3.2.3.2. Here we are working over a fixed pseudorepresentationinto a complete local ring, and the condition ΦD is the finitude of the vector space of self-extensions of thesemisimple representation corresponding to the pseudorepresentation of the special fiber D. This shows thatthe complete local ring can be taken to be Noetherian (Theorem 3.1.5.3), but finite generation of the algebraover this base is not required. The condition on the self-extensions suffices.

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Theorem 1.2.2.4 (Dubnov-Ivanov [DI43]). Let k be a field and let R be a nil k-algebrasuch that there exists a positive integer d with the property that rd = 0 for every r ∈ R. Thenif char(k) = 0 or char(k) > d, there exists some N ≤ 2d − 1 such that RN = 0.

There exist examples showing that 2d − 1 is the best possible such bound.

Remark 1.2.2.5. This theorem is known as the Nagata-Higman theorem, since it wasdiscovered in the western mathematical community by Nagata [Nag52] in characteristic zero,and then generalized to large enough positive characteristic by Higman [Hig56]. It was firstdiscovered Dubnov and Ivanov [DI43] but overlooked in the west. For a few further remarkson the history and context of these works, see [For90].

The following theorem is more in the spirit of Shirshov’s height theorem [Sir57], andfulfills Chenevier’s suspicion [Che11, Remark 2.29] that there exists some such result whichwill allow one to show the nilpotence of the kernel of a Cayley-Hamilton pseudorepresentationover a field, even when the characteristic is too small to apply the Nagata-Higman theorem.For further comments on Shirshov’s height theorem, see [Kem09].

Theorem 1.2.2.6 (Samoilov [Sam09]). Let R be an associative PI algebra over a field kof characteristic p > 0. If R is generated by a set X and every word in the elements of x isnilpotent of degree not exceeding d, then R is a nilalgebra, i.e. there exists a positive integerN such that RN = 0. Here N depends on p, the particular polynomial identity it satisfies,and on d, but it does not depend on the cardinality of X.

For future reference, let us record a particular integer N = N(p, d).

Definition 1.2.2.7. Let p be a prime number and let d be a positive integer. LetN(p, d) be the integer determined by Theorem 1.2.2.6, where, in the notation of the theoremstatement, p is the characteristic of the field k, d is the bound on the nil-degree of theelements of X, and the polynomial identity is xd. Let N(d) be the integer specified inCorollary 1.2.2.8 below.

For a fixed d, Theorems 1.2.2.4 and 1.2.2.6 combine to form the following result.

Corollary 1.2.2.8. There exists an integer N(d) ≥ 0 dependent only on d with thefollowing property: for any associative, non-unital algebra R over a characteristic p ≥ 0 fieldk such that every element of R satisfies the identity xd where d ≥ 1, R is nilpotent of degreeno more than N(d), i.e. RN(d) = 0. The integer N(d, p) also has this property over suchalgebras R where k has characteristic p.

Proof. Let N(d) be the maximum of the finite collection of integers

N(p, d) : prime p ≤ d ∪ 2d − 1.Then by Theorems 1.2.2.4 and 1.2.2.6, RN(d) = 0.

While we will prove stronger results later, let us now list some immediate corollaries,applying the results from PI theory above to Cayley-Hamilton A-algebras.

Corollary 1.2.2.9. Let (R,D) be a finitely generated Cayley-Hamilton A-algebra ofdegree d, where A is a commutative Noetherian ring. Then R is finite as a module over A.

Proof. As (R,D) is a Cayey-Hamilton A-algebra, each element r ∈ R satisfies its char-acteristic polynomial, which is a degree d monic polynomial equation χ(r, t) with coefficients

46

in A. Proposition 1.2.2.2 implies that R is a PI A-algebra, and then Theorem 1.2.2.3 impliesthat R is finite as a module over A.

There are several more very useful consequences of this finiteness, which we now discuss.

Corollary 1.2.2.10. Let (R,D) be a d-dimensional finitely generated Cayley-HamiltonA-algebra, where A is a commutative Noetherian ring.

(1) R is finite as an A-module; in particular, it is finite over its center and is a Noe-therian ring.

(2) ker(D) ⊂ R is a nilpotent two-sided ideal.(3) If A is a Jacobson ring (e.g. a field), then R is a Jacobson ring as well, and J(R) =

N(R) is an equality of nilpotent ideals.(4) If A is an Artinian ring, then R is as well.

Proof. The first statement in (1) repeats Corollary 1.2.2.9. When A is a commutativeNoetherian ring, then an A-algebra which is finite as an A-module via the structure map isalso Noetherian (see e.g. [MR01, Lemma 1.1.3]). This proves (1).

Because (R,D) is Cayley-Hamilton, each element r ∈ ker(D) satisfies its characteristicpolynomial χ(r, t) = td. Therefore the kernel is a nil two-sided ideal. Since R is Noetherian,the nilradical of R contains ker(D) and is nilpotent (see Remark 1.2.2.11 below). Henceker(D) is nilpotent as well.

If A is a Jacobson ring, then R is a Jacobson ring as well, as it is finite as a moduleover A [MR01, §9.1.3] (see also [MR01, Theorem 13.10.4(iii)]). Therefore its nilradical isthe same as its Jacobson radical. As R is Noetherian, both are nilpotent (see the Remarkimmediately below). This proves (3).

Taking R as an A-module, it is the descending chain condition holds on sub-A-modules ofR because it is a finitely generated module over an Artinian ring. As ideals of R are certainsub-A-modules of R, the descending chain condition also holds for ideals, proving (4).

Remark 1.2.2.11. There are several notions of nilradical which coincide for Noetherianrings. Here are the notions for a general noncommutative ring R.

(1) The lower nilradical is the intersection of all prime ideals in a ring, where an idealI ⊂ R is prime if for any ideals A,B such that A · B ⊆ I, then either A ⊆ I orB ⊆ I.

(2) The Levitsky radical is the largest locally nilpotent ideal, where an ideal is calledlocally nilpotent if any finitely generated sub-ideal is nilpotent.

(3) The upper nilradical is the ideal generated by all nil ideals in R, where an ideal iscalled nil if every element in it is nil. Note that the ideal generated by nilpotentelements may not be nil in the noncommutative case; this definition of upper radicalis chosen so that it the upper radical is a nil ideal.

In general there is an inclusion

lower nilradical ⊆ Levitsky radical ⊆ upper nilradical,

but one can check that these definitions coincide when R is Noetherian, so that one canspeak of “the nilradical of R.” In particular, in the Noetherian case, its follows from thisequivalence that the nilradical is a nilpotent ideal.

The Jacobson radical always contains the (upper) nilradical, and is equal to the nilradicalwhen R is Jacobson and Noetherian. For more information see e.g. [GW04].

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1.2.3. The Jacobson Radical of a Cayley-Hamilton Algebra. We write J(R) forthe Jacobson radical of an algebra R.

The following lemma is a strengthening of a lemma of Chenevier [Che11, Lemma 2.7-2.8], beginning an exploration of the extent to which the kernel of a d-dimensional Cayley-Hamilton pseudorepresentation behaves like a nilpotent subalgebra (without unit) of a matrixalgebra. The addition to and partial simplification of Chenevier’s arguments comes from PIring theory.

Lemma 1.2.3.1 (Following [Che11, Lemma 2.7]). Let D : R −→ A be a Cayley-Hamiltonpseudorepresentation, where A is a commutative ring and R is an A-algebra.

(1) J(R) is the largest two-sided ideal J ⊂ R such that D(1 + J) ⊂ A×.(2) For any r ∈ ker(D), we have (rr′)d = 0 for all r′ ∈ R. In particular, ker(D) is a nil

ideal and is contained in the upper nilradical of R, and therefore also contained inJ(R).

Now assume that A is a field.

(3) r ∈ R is nilpotent if and only if D(t− r) = td. Moreover, J(R) consists of nilpotentelements.

(4) ker(D) and J(R) are nilpotent ideals, with degree of nilpotence bounded by the integerN(d) of Definition 1.2.2.7, which depends only on the integer d. However, if d! isinvertible in A, then the bound 2d − 1 suffices.

(5) ker(D) = J(R).(6) If I ⊂ R is a two-sided ideal such that In = 0 for some n ≥ 1, then I ⊂ ker(D)

(here it is not necessary to assume that D is Cayley-Hamilton).

Remark 1.2.3.2. This lemma and its proof is based on Chenevier’s lemma [Che11,Lemma 2.7]. It is due to him, except for (3), which comes from our use of PI ring theory.

Proof. Without applying a Cayley-Hamilton assumption, if r ∈ R is invertible, thenD(r) is invertible since D is multiplicative and preserves units. Assuming the Cayley-Hamilton property, the converse is true: if D(r) = a is invertible in A, then the multiplicativeinverse of r is given by manipulating its characteristic polynomial.

(1.2.3.3) (rd−1 − Λ1(r)rd−2 + · · ·+ (−1)d−1Λd−1(r)) · r = −a.Since the Jacobson radical J(R) of R is the set of quasiregular elements, i.e. r ∈ R such that1− r is a unit in R, we see that r ∈ J(R) if and only if D(1− r) is a unit, proving (1).

Now we will prove (2). If r ∈ ker(D) and r′ ∈ R, then Λi(rr′) = 0 for 1 ≤ i ≤ d. Then r

must satisfy the characteristic polynomial χ(r, t) = td. This shows that ker(D) is a nil idealof bounded nil-degree d. Therefore ker(D) ⊆ N(R).

Now let A be a field k. If r ∈ R is nilpotent, then 1 + tr ∈ R ⊗k k[t] is invertible.Therefore D(1 + tr) is invertible in k[t], hence D(1 + tr) is in k×. Using the homogenousmultiplicativity of D on B = A[t, t−1], we see that

t−d ·DB(1− tr) = DB(t−1 − r) = χ(r, t−1),

so that χ(r, t) = td and D(1 + tr) = 1. Therefore rd = 0, proving one direction of part (3).For the converse, we simply use the Cayley-Hamilton property. Now choose x ∈ J(R). Forall y ∈ k[x], 1 + yx is invertible in k[x], so that D(1 + yx) ∈ k×. Then, as in the proofof part (1), we know that 1 + yx is invertible in k[x]. This means that x ∈ J(k[x]). Thisonly happens when k[x] is finite dimensional as a k-vector space. Since any element of the

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Jacobson radical of a finite dimensional algebra over a field is nilpotent, we conclude that xis nilpotent as desired. This concludes (3).

Parts (2) and (3) have shown that ker(D) and J(R) are nil-ideals of bounded nil-degreed, i.e. all of their elements are nilpotent of degree d. Part (4) follows directly from this fact,upon applying Corollary 1.2.2.8.

To prove (5), let us first assume that k is an infinite field. We know from part (3) thatJ(R) consists of nilpotent elements, and that D(1 + r) = 1 for all r ∈ J(R). Since k is aninfinite domain and J(R) is a two-sided ideal, we may apply Lemma 1.1.7.2, which tells usthat

ker(D) = r ∈ R | ∀r′ ∈ R,D(1 + rr′) = 1.This shows that J(R) ⊆ ker(D). The opposite inclusion is part (2). It remains only toreduce to the case that k is an infinite field; this is accomplished in Lemma 1.2.3.5 below.This completes our proof that J(R) = ker(D) when A is a field.

For part (6), let I be a nilpotent ideal of R and choose r ∈ I. Then for any y ∈R⊗AA[t1, · · · ,m] for any m, ry is nilpotent. Therefore D(1 + try) is invertible, hence equalto 1 by the logic above. Therefore r ∈ ker(D) by definition.

Remark 1.2.3.4. The nilpotence of the nilradical of a finitely generated PI algebra overa commutative Noetherian ring was first proved by Braun [Bra84]. We proved this moresimply because, in our case of concern, R is integral of bounded degree over A and thereforefinite as an A-module.

Lemma 1.2.3.5 ([Che11, Lemma 2.8]). Let k be a field and let D : R → k be a d-dimensional pseudorepresentation. Then for any separable algebraic extension K/k, the nat-ural injection R⊗k K induces isomorphisms

J(R)⊗k K∼−→ J(R⊗k K), ker(D)⊗k K

∼−→ ker(D ⊗k K).

This proof is due to Chenevier.

Proof. By Lemma 1.1.5.2, we have an injection

ker(D)⊗k K −→ ker(D ⊗k K).

We need to show that this map is surjective. Enlarge K if necessary, so that K/k is normalwith Galois group Γ. Consider the natural semilinear action of Γ on R ⊗k K. By Hilbert’sTheorem 90, each Γ-stable K-subvector space of V of R⊗kK has the form V Γ⊗kK, whereV Γ ⊂ R is the k-vector space of fixed points. We claim that ker(D ⊗k K) is Γ-stable.Observe that Γ has a natural semilinear action on any K-algebra B. As the characteristicpolynomial coefficient functions of D⊗kK are defined over k, we have for any K-algebra B,any r ∈ R⊗k B, and any γ ∈ Γ that D is Γ-equivariant, i.e.

D(γ(r)) = γ(D(r)).

The claim now follows upon examining the definition of the kernel: if r ∈ ker(D⊗kK), thenD(1+rr′) = 1 for all K-algebras B and r′ ∈ R⊗kB, and this will remain true after replacingr with σ(r). Now the desired surjectivity follows from the fact that ker(D⊗kK)Γ ⊂ ker(D).This also follows from the Γ-equivariance of D.

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1.2.4. The Universal Cayley-Hamilton Algebra. This paragraph discusses a trivialgeneralization of [Che11, 1.22-1.23], introducing the category of “Cayley-Hamilton represen-tations” of a given A-algebra R. We may think of this as a generalization of the universalAzumaya-algebra valued representation of R discussed in §1.4 below.

We start with the usual data of an algebra R over a commutative ring A. From Theorem1.1.7.4, we have the universal pseudorepresentation

Du : R⊗A ΓdA(R)ab −→ ΓdA(R)ab

of R over ΓdA(R)ab. Now we apply the notion of a Cayley-Hamilton algebra to this universalpseudorepresentation.

Definition 1.2.4.1. Let R,A, and Du be as above. Let B a commutative A-algebra.

(1) A Cayley-Hamilton B-representation of R of dimension d over B is a triple

(B, (E,D), ρ)

where (E,D) is a Cayley-Hamilton A-algebra relative to the pseudorepresentationD : E → B, and ρ : R⊗A B → E is a homomorphism of B-algebras.

(2) The universal Cayley-Hamilton representation of R is

(ΓdA(R)ab, (E(R, d), Du|E), ρu),

where E(R, d) is the ΓdA(R)ab-algebra

E(R, d) := (R⊗A ΓdA(R)ab)/CH(Du)

receiving the canonical quotient homomorphism ρu : R⊗A ΓdA(R)ab → E(R, d), andDu|E : E(R, d)→ ΓdA(R)ab is the factorization of Du through ρu.

Of course, the factorization Du|E exists, in view of Lemma 1.1.6.6(2) and Lemma 1.2.1.1.

Remark 1.2.4.2. Cayley-Hamilton representations are direct generalizations of the usualnotion of a representation. With R,A as usual, let R ⊗A B → Md(B) be a B-valued d-dimensional representation of R. Then

(B, (Md(B), det), ρ)

is a d-dimensional Cayley-Hamitlon representation of R over B, where det is the standarddeterminant map det : Md(B)→ B.

We want to show that the “universal” d-dimensional Cayley-Hamilton representation ofR deserves its name, but first we must define the structure of a category CHd(R) wherethis representation will be initial, following [Che11, §1.22]. The objects are the data of thedefinition above, and a morphism of Cayley-Hamilton representations of R

(B1, (E1, D1), ρ1) −→ (B2, (E2, D2), ρ2)

is a pair (f, g) where f : B1 → B2 and g : E1 → E2 are ring homomorphisms such that ifιi : Bi → Ei is the Bi-algebra structure on Ei, then the diagrams

B1ι1 //

f

E1

g

E1D1 //

g

B1

f

B2ι2 // E2 E2

D2 // B2

50

and

R⊗A B1ρ1 //

id⊗f

E1

g

R⊗A B2

ρ2 // E2

commute.

Proposition 1.2.4.3 ([Che11, Proposition 1.23]). The universal d-dimensional Cayley-Hamilton representation

(ΓdA(R)ab, (E(R, d), Du|E), ρu)

is the initial object of CHd(R).

Proof. Let (B, (S,D), η) be a d-dimensional Cayley-Hamilton representation of R. TheB-algebra homomorphism η : R ⊗A B → S induces a d-dimensional B-valued pseudorepre-sentation of R, namely D η. This induces an A-algebra homomorphism f : ΓdA(R)ab → B.This in turn induces an A-algebra homomorphism

R⊗A ΓdA(R)ab −→ R⊗A Bη−→ S.

Since (S,D) is Cayley-Hamilton, Lemma 1.1.8.6 implies that this map factors through ρu :R⊗A ΓdA(R)ab E(R, d), with quotient

g : E(R, d) −→ S.

We observe that f Du|E = Dg, and that (f, g) has the remaining properties of a morphismin CHd(R), as desired.

Now, assuming that A is Noetherian and R is finitely generated as an A-module, we havea pleasant consequence of the PI theory of §1.2.2. This proposition will be applied in §1.4.3to show that the representation theory of such an algebra R reduces to the representationtheory of a finite-over-center algebra, basically by exploring the consequences of Remark1.2.4.2.

Proposition 1.2.4.4. If A is Noetherian and R is finitely generated as an A-algebra,then the universal d-dimensional Cayley-Hamilton algebra of degree d associated to R, namelythe ΓdA(R)ab-algebra E(R, d), is finite as a ΓdA(R)ab-module. In particular, E(R, d) is aNoetherian ring and is finite as a module over its center.

Proof. This is an instance of Corollary 1.2.2.10(1).

1.3. Pseudorepresentations over Fields

In the current chapter, we are developing the theory of pseudorepresentations and then,starting in §1.4, studying the moduli space of pseudorepresentations relative to the modulispace of representations. The main theorem of this chapter, Theorem 1.5.4.2, depends heavilyon the comparison of representations wtih pseudorepresentations over an algebraically closedfield. Indeed, it is fair to say that in Chapter 1 we prove what we can about this situationby studying moduli functors through their geometric points, and in Chapter 2 we aim for acloser, local-on-the-base study.

This is our motivation for studying pseudorepresentations over fields. We can find a closerelationship between semisimple representations and pseudorepresentations over fields. We

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will now give the main theorem. This is a critical property of this notion of pseudorepresen-tation, developed by Chenevier, who calls it a “determinant.” Previous notions, which wecall “pseudocharacters” here, did not function well in the case that the dimension is greaterthan or equal to the characteristic of the field (see §1.1.12).

1.3.1. Main Theorem. As usual, let R is an A-algebra. We have seen that represen-tations of R valued in an Azumaya algebra, and in particular a matrix algebra, induce apseudorepresentation (Theorem 1.1.7.4(6)). We have also seen that given a pseudorepre-sentation D of R, the universal Cayley-Hamilton representations of R over D share somesimilarities with representations of R valued in subalgebras of matrix algebras (cf. Corollary1.2.2.10). Now we will show that over an algebraically closed field, pseudorepresentationsare in natural bijection with representations. Here is our main theorem, due to Chenevier.

Theorem 1.3.1.1 ([Che11, Theorem 2.12]). Let k be an algebraically closed A-field.There is a bijection between conjugacy classes of semisimple d-dimensional representationsρ of R over k and d-dimensional pseudorepresentations of R over k, given by sendingρ : R ⊗A k → Md(k) to det ρ. In fact, if D is a d-dimensional k-valued pseudorepre-sentation of R, then the corresponding semisimple representation may be written as

R⊗A k −→ (R⊗A k)/ ker(D) '∏

Mdi(k),

where∑di = d.

We will also find an analogous result over arbitrary A-fields k. Indeed, Theorem 1.3.1.1follows directly from this more general case. However, it will require that we establish somenotions and notation.

The following notion of an “exponent” describes the size of field extensions K/k in adifferent way than the degree of an extension. Indeed, the exponent may be finite andmeaningful even when the degree of the extension is infinite. We also give “determinant”maps from central simple algebras S/K to k ⊂ K when K/k has finite exponent, generalizingthe determinant on a matrix algebra.

Definition 1.3.1.2. Let K/k be a field extension, and let k′ ⊂ K be the maximalseparable subextension of K. Assume that k′/k is finite. If the characteristic p of k ispositive, let q be the smallest power of p such that Kq ⊆ k′, and if p = 0 let q = 1. Definethe exponent (f, q) ∈ N2 of K/k by f = [k′ : k] and q as above. It is possible for both oreither of the quantities in the exponent to be infinite.

Now assume that K/k has finite exponent (f, q), and let S be a central simple K-algebraof rank n2 over K with its reduced norm N : S → K. Let Nk′/k : k′ → k be the norm mapon finite separable fields, and let F q : K → k′ be the q-power Frobenius map. Then there isa natural determinant

detS : S → k

of k-homogenous degree nqf defined by detS = Nk′/k F q N .

We observe that in the case that the exponent of S is trivial (1, 1), detS is the standardreduced norm of an Azumaya algebra, such a matrix algebra.

Now we can state the theorem describing pseudorepresentations of an algebra over anarbitrary field

Theorem 1.3.1.3 ([Che11, Theorem 2.16]). Let R be a k-algebra. Let D : R → k be ad-dimensional pseudorepresentation.

52

(1) Then there is an isomorphism of k-algebras

R/ ker(D)∼−→

s∏i=1

Si

where Si is a simple k-algebra which is of finite dimension n2i over its center ki, and

where ki/k is a with finite exponent (fi, qi). In particular, R/ ker(D) is semisimple.(2) Moreover, under such an isomorphism, D is equal to the sum of determinants

D =s⊕i=1

detmiSi , d =s∑i=1

miniqifi,

where mi are certain uniquely determined integers.(3) The pseudorepresentation D is realizable as the composition of the natural sum of

determinants detSi with the following product of the natural surjections R Si,namely

R −→s∏i=1

mi∏j=1

Si,

where the integers mi are as above.(4) R/ ker(D) is finite-dimensional as a k-vector space and, equivalently, each ki is

finite-dimensional if any of the following conditions are satisfied, where p is thecharacteristic of k.(a) k is perfect,(b) d < p,(c) p > 0 and [k : kp] <∞, or(d) R is finitely generated as a k-algebra.

Let us deduce the algebraically closed case from this general case.

Proof. (Theorem 1.3.1.3 implies Theorem 1.3.1.1.) Beginning with the notation ofTheorem 1.3.1.1, we let k be an algebraically closed A-field and replace R by R ⊗A k andthink of R as a k-algebra and let D be a d-dimensional pseudorepresentation D : R→ k.

By definition of the exponent, every element of a field extension K/k of finite exponentis algebraic over k. Since k is algebraically, closed this means that K = k when K/k hasfinite exponent, i.e. the exponent is (1, 1). Now Theorem 1.3.1.3 implies that R/ ker(D)is a product of central simple k-algebras, which are therefore matrix algebras because k isalgebraically closed. We write

R/ ker(D)∼−→

s∏i=1

Mdi(k).

If we write deti for the determinant function on Mdi(k), Theorem 1.3.1.3 tells us that

D ∼=s⊕i=1

detmii , wheres∑i=1

midi = d,

53

and where ⊕ refers to the direct sum of (1.1.11.2). If Vi a the di-dimensional representationof R corresponding to R Mdi(k), then clearly the representation

s⊕i=1

V ⊕mii

realizes D as its determinant. Since, by the Brauer-Nesbitt theorem, a semisimple represen-tation over an algebraically closed field is determined up to isomorphism by its character-istic polynomials, this semisimple representation is unique up to isomorphism. Conversely,a pseudorepresentation is determined by its characteristic polynomial functions (Corollary1.1.9.15). Therefore the correspondence is bijective.

We will prove Theorem 1.3.1.3 in the next paragraph.

1.3.2. Semisimple k-algebras. Now we work toward proving Theorem 1.3.1.3. Firstly,we will note that our existing knowledge allows us to conclude immediately that R/ ker(D)is semisimple and track the number of orthogonal idempotents.

Recall that R is a k-algebra with a d-dimensional pseudorepresentation D : R→ k. Letp be the characteristic of k.

Because Lemma 1.2.1.1 tells us that (R/ ker(D), D) is a Cayley-Hamilton k-algebra, wecan apply our study of Cayley-Hamilton algebras from §1.2. Let us review the facts that wecan deduce directly from this study.

• Every element of R is integral (i.e. algebraic) of bounded degree d over k: eachelement satisfies its own characteristic polynomial.• By Proposition 1.2.2.2 R is a PI-k-algebra.• By Lemma 1.2.3.1(5), ker(D) is the Jacobson radical J(R) of R, so R/ ker(D) is

semisimple.• By Lemma 1.2.1.4, the largest possible cardinality of a family of pairwise orthogonal

idempotents of R/ ker(D) is d.

Also, Corollary 1.2.2.9, if R is finitely generated as a k-algebra, R/ ker(D) is a finite dimen-sional k-algebra. However, we are not currently assuming that R is finitely generated as ak-algebra.

All that we need to do is to control the exponent of the centers of the simple fac-tors (Lemma 1.3.2.1 below) and control the possible pseudorepresentations out of simplek-algebras (Lemma 1.3.2.3 below).

The following lemma describes field extensions of k satisfying the first property of thebullet list above; these are the possible fields that can appear as the center of a k-algebrasatisfying all of the properties of the bullet list.

Lemma 1.3.2.1 ([Che11, Lemma 2.14]). If S is a k-algebra satisfying the properties inthe bullet list above, then

S∼−→

s∏i=1

Mni(Ei)

where Ei is a division k-algebra, finite dimensional over its center ki, and s ≤ d. In partic-ular, S is semisimple. The center ki of Ei is a finite separable extension of k, unless k haspositive characteristic p, in which case k[kqi ] is separable, where q is the greatest power of pless then n. Moreover, S is finite dimensional over k if any of the following conditions aresatisfied:

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(1) k is perfect,(2) p > d,(3) p > 0 and [k : kp] <∞, or(4) R is finitely generated over k.

We record some of the proof here for reference, following the proof of [Che11, Lemma2.14].

Proof. Let A be a commutative k-algebra satisfying the properties in the bullet list. Ifp > 0, define q as in the statement of the lemma, and set q = 1 otherwise. The bound onthe number of idempotents implies that

A∼−→

s∏i=1

Ai

where s ≤ d, and where Ai is an algebraic field extension of k. Since Ai/k has boundedalgebraic degree d, its maximal separable subextension Aet

i is finite dimensional over k. Asthe center Z(S) of S has the properties of A, we have established the conditions of the lastpart of the lemma are sufficient to imply that the center is finite dimensional over k.

Now we show that S is semisimple. Let M be a simple S-module, and let E be thedivision algebra EndS(M). First, we claim that M is finite dimensional over E. Indeed,Jacobson’s density theorem6 implies that either M is finite dimensional over E and S →EndE(M) ' Ms(E

op) is surjective, or for each j ≥ 1 there is a k-subalgebra Rj ⊂ S and asurjective k-algebra homomorphism Rj Mj(E

op), but the second option is not possiblesince the elements of S are algebraic of bounded degree over k.

Now we claim that there are finitely many simple S-modules M1, . . . ,Ms up to isomor-phism. This will complete the proof that S is semisimple, for in this case the fact thatJ(S) = 0 implies that

(1.3.2.2) S −→s∏i=1

Mni(Eopi ), where Ei := EndS(Mi)

is injective, and the fact that the Mi are pairwise non-isomorphic implies that it is surjective.It remains to check the claim. For this, we refer the reader to the remainder of the proof,

found in [Che11, Lemma 2.14].

Now we must describe the possible pseudorepresentations out of a simple k-algebra Swhose center K is a finite exponent extension of k. Let us first recall that we have alreadygiven such a result in the case that the center of S is k, so that S is an Azumaya algebraover k. This is Proposition 1.1.7.10, due to Ziplies [Zip86], which states that all of thepseudorepresentations out of an Azumaya algebra are induced by integral powers of thereduced norm.

Having described the Azumaya algebra case, we proceed to the general case.

Lemma 1.3.2.3 ([Che11, Lemma 2.17]). Let K/k be a field extension with finite exponent(f, q), and let S be a central simple finite dimensional K-algebra. Then any pseudorepresen-tation D : S → k has the form detmS for some unique integer m ≥ 0.

6Jacobson’s density theorem states that for any simple left module N of a ring R, any EndR(N)-lineartransformation η of N , and any finite set of elements xi of N , there exists r ∈ R such that η(xi) = r · xifor all i. See e.g. [Her68, Theorem 2.1.2].

55

Proof. Let D : S → k be a d-dimensional pseudorepresentation, and define n2 :=dimK(S). Note that if D = detmS , then we must have d = fmnq since detS is homogenousof degree fnq by definition; this shows that m is unique if it exists.

We will use the following fact below: if two d-dimensional pseudorepresentations D1, D2 :R→ A are such that D1 ⊗A B ∼= D2 ⊗A B for some commutative A-algebra B with A→ Binjective, then D1

∼= D2. This follows directly from the representability of the moduli spaceof pseudorepresentations, Theorem 1.1.7.4.

Assume for the moment that k is separably closed, so that K is as well. The Noether-Jacobson theorem implies that S is isomorphic to some matrix algebra Mn(K), n ≥ 1. SetA := K⊗kK, and denote by I the kernel of the natural split surjection A→ K. We see thatI is generated as an A-module by elements of the form x⊗ 1− 1⊗ x, which are nilpotent ofindex ≤ q. In, particular, I is a nil ideal, and any finite type A-submodule of I is nilpotent asan ideal. Now Lemma 1.2.3.1(6) implies that for any pseudorepresentation D : Mn(A)→ K,D factors through π : Mn(A) Mn(A/I) = Mn(K). Applying this to

D ⊗k K : S ⊗k K 'Mn(K)⊗k K ∼= Mn(A) −→ K,

we get a pseudorepresentation Mn(K) → K, which is an integral power of the usual deter-minant by Proposition 1.1.7.10, say D ⊗k K ∼= detsMn(K) π and d = ns. Now recall thatthe restriction of D ⊗k K to Mn(K) ⊗ 1 ⊂ Mn(A) must be valued in k, since D is valuedin l. This means that dets(Mn(K)) ⊂ k. Therefore q must divide s, and we observe that

dets/qS ⊗kK ∼= D ⊗k K. Now by the fact mentioned above, this implies that D ∼= det

s/qS .

Now we reduce to the case that k is separably closed. We have

K ⊗k ksep ∼−→f∏i=1

Ki,

where Ki = K · ksep is a separable algebraic closure of K such that Kqi ⊂ ksep for each i (q

is minimal for this property) and Gal(ksep/k) permutes transitively the Ki. Recall that f isthe (finite) separable degree of K over k. Likewise,

S ⊗k ksep ∼= S ⊗K (K ⊗k ksep)∼−→

f∏i=1

Si,

where Si = S ⊗K Ki is central simple of rank n2 over Ki. By Proposition 1.1.11.5(2), each

D ⊗k ksep is the product of determinants Si∼→ Mn(Ki) → ksep, which have the form detmiSi

by the previous step above, and d = n(∑f

i=1mi). As D ⊗k ksep is Gal(ksep/k)-equivariant,this implies that mi is independent of i, i.e. mi = m for each i. Therefore, m = d/nf , andwe observe that D ⊗k ksep ∼= detmS ⊗kksep. Now by the fact mentioned above, this impliesthat D ∼= detmS , as desired.

Now we complete the proof of Theorem 1.3.1.3.

Proof. (Theorem 1.3.1.3) By Lemma 1.3.2.1, we know that R/ ker(D) is isomorphic toa product of s ≤ d simple k-algebras Si whose centers ki are finite exponent extensions of k.This is part (1). Write (fi, qi) for the exponent of ki.

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By Proposition 1.1.11.5(2), any pseudorepresentation out of R/ ker(D) is the sum7 ofpseudorepresentations Di, one out of each Si. Indeed, Spec k is connected, so that the con-ditions of Proposition 1.1.11.5 are satisfied. Lemma 1.3.2.3 implies that each Di is a powerdetmiSi of detSi . As Proposition 1.1.11.5 tells us that the degree of a sum of pseudorepresen-tations is the sum of the degrees, and detSi has degree niqifi, the formula for the degreefollows; this is part (2).

For part (3), we are simply combining part (1) with Lemma 1.1.6.6(2). Part (4) followsdirectly from Lemma 1.3.2.1.

Corollary 1.3.2.4. Let D : R → k be a d-dimensional pseudorepresentation of a k-algebra R.

(1) There exists a field extension L/k such that D⊗k L is realizable as the determinantof a matrix algebra-valued representation

R⊗k K →Md(L).

If R/k is finitely generated, then L/k may be chosen to be a finite extension.(2) When the centers ki/k, 1 ≤ i ≤ s, of exponent (fi, qi), the simple factors Si of

R/ ker(D) of Theorem 1.3.1.3 are separable extensions, e.g. when k is perfect, thenthere exists a finite separable extension K of degree bounded by

∏si=1 fi such that

R/ ker(D)⊗k Kis a product of matrix algebras and the natural map from R⊗k K to this algebra isa d-dimensional representation whose determinant induces D ⊗k K.

Proof. We begin with the case that the integer s from Theorem 1.3.1.3 is 1, i.e. thek-algebra R/ ker(D) is a central simple n2-dimensional L-algebra S where L/k is a fieldextension of finite exponent (f, q) such that d = n ·f · q. Its maximal separable subextensionL′/k has degree f . Because universal homeomorphisms such as inseparable extensions induceequivalences of etale topoi and Brauer groups classify central simple algebras over a field,the L′ algebra S ⊗k L′ is isomorphic to Mn(L). We then observe that the product L-algebra

q∏i=1

∏σ∈Gal(L′/k)

σMn(L)

is naturally embeddable in Md(L). The pseudorepresentation resulting from

R/ ker(D)⊗k L −→q∏i=1

∏σ∈Gal(L′/k)

σMn(L) −→Md(L)det−→ L

is then equal to D ⊗k L, upon examining the “determinant” detS of S defined in Definition1.3.1.2 and the conclusion of Theorem 1.3.1.3.

The general result (1) follows by applying this to each of the simple factors Si ofR/ ker(D), taking the sum of the resulting pseudorepresentations on the product of thesefactors, and tensoring D by the composite field of the extensions L′ above of each factor.

The claim that the finite generation of R/k implies the finitude of L/k follows fromTheorem 1.3.1.3(4d).

Part (2) follows from part (1) and its proof when qi = 1 for each i.

7Recall that this sum is defined in (1.1.11.2).

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1.3.3. Finite-Dimensional Cayley-Hamilton Algebras. In this paragraph, we findconditions under which a Cayley-Hamilton algebra (R,D) over a field k is finite-dimensional.There are three basic ingredients. Results from PI ring theory from §1.2.2 culminated in thefact that the Jacobson radical of a Cayley-Hamilton algebra over a field is nilpotent, withdegree of nilpotence bounded in terms of the degree of the pseudorepresentation (Lemma1.2.3.1). The next ingredient is the conditions we have given above for the maximal semisim-ple quotient of R to be finite-dimensional. Finally, we require some basic lemma, which wenow give. This translates the condition that ker(D)/ ker(D)2 is a finite dimensional vectorspace, which is the last fact we require, into a condition on the deformations to k[ε]/(ε2) ofthe semisimple representation ρ associated to D.

Lemma 1.3.3.1. Let A be a commutative ring, R an A-algebra. Let I be a two-sided idealof R. There is a natural A-module isomorphism

HomR(I/I2, R/I)∼−→ Ext1

R(R/I,R/I).

Proof. Apply HomR(−, R/I) to the exact sequence of R-modules

0 −→ I −→ R −→ R/I −→ 0,

and use that Ext1R(R,−) = 0.

We continue to work with deformations of a fixed d-dimensional pseudorepresentationD : R → k. Now let us restrict to the case that S := R/ ker(D) is finite dimensional as ak-vector space.

Theorem 1.3.3.2. Let k be a field of characteristic p ≥ 0 and let R be a k-algebraequipped with a Cayley-Hamilton d-dimensional pseudorepresentation D : R → k. Assumethat S := R/ ker(D) is finite-dimensional over k. If Ext1

R(S, S) is finite-dimensional as a k-vector space, where S is treated as an R-module here, then R is finite-dimensional k-algebra.

Recall that sufficient conditions for S to be finite dimensional over k are given in Theorem1.3.1.3(4).

Proof. Apply Lemma 1.3.3.1, so that the assumption that dimk Ext1R(S, S) <∞ implies

that dimk HomS(ker(D)/ ker(D)2, S) < ∞. This means that ker(D)/ ker(D)2 is a finitesum of simple representations of S, but this in turn implies that ker(D)/ ker(D)2 is finite-dimensional as a k-vector space.

Because there are natural surjections

(I/I2)⊗nk In/In+1

for any ideal I ⊂ R, this means that R/ ker(D)2n is also finite dimensional over k for anypositive integer n. Since (R,D) is Cayley-Hamilton, Lemma 1.2.3.1(4) implies that ker(D) isnilpotent of index bounded by N(d) (or by N(d, p)), where N(d) is the integer of Definition1.2.2.7. This completes the proof.

1.3.4. Composition Factors of Field-Valued Pseudorepresentations. We con-clude this section with some discussion of the simple factor algebras of R/ ker(D), wherewe continue to let R be a k-algebra where k is a field. Equivalently (almost), we discussthe Jordan-Holder factors that appear in representations of R arising from pseudorepresen-tations according to Theorem 1.3.1.1. We mostly follow Chenevier’s discussion of [Che11,§2], and introduce some notions – the Grothendieck group of R and dimension vectors of

58

representations – that will be useful in §2.2. These notions will be heavily used when wediscussion deformation theory of pseudorepresentations in Chapter 2.

Definition/Lemma 1.3.4.1 ([Che11, Defn.-Prop. 2.18]). Let D : R → k be a d-dimensional pseudorepresentation over a field k. We call D absolutely irreducible providedthat one of the following equivalent conditions is true.

(1) The semisimple representation ρD : R⊗k k →Md(k) with determinant equal to thepseudorepresentation D, which exists and is unique up to isomorphism by Theorem1.3.1.1, is irreducible,

(2) (R⊗k k)/ ker(D ⊗k k) 'Md(k),(3) R/ ker(D) is a central simple k-algebra of rank d2,(4) R/CH(D) is a central simple k-algebra of rank d2,(5) for some (resp. all) subset X ⊂ R generating R as a k-vector space, there exists

x1, x2, . . . , xd2 ∈ X such that the abstract d2 × d2 matrix ((Λ1(xixj))i,j belongs toGLd2(k).

If they are satisfied, then CH(D) = ker(D) = x ∈ R, ∀y ∈ R,Λ1(xy) = 0.

Proof. Since we know from Proposition 1.1.7.10 that any pseudorepresentation out of amatrix algebra is a power of the determinant, and a pseudorepresentation factors through thequotient by its kernel, (2) implies (1) since we know that D has dimension d. Conversely, if ρ :R⊗k k →Md(k) is as in (1), then Wedderburn’s theorem tells us that ρ is surjective. We seethat ker(ρ) ⊂ ker(D), since the pseudorepresentation det ρ is invariant under multiplicationby ker(ρ). Therefore (2) follows from Theorem 1.3.1.3. Also, (5) (for any subset X ⊂ Rsatisfying the conditions above) follows from (1) or (2) by the nondegeneracy of the tracepairing on Md(k). Conversely, if X ⊂ R satisfies (5), then

dimk((R⊗k k)/ ker(D ⊗k k)) ≥ d2,

and now (5) implies (2) by Theorem 1.3.1.3, since positive integers ni, 1 ≤ i ≤ s, such that∑s1 ni = d also satisfy

∑n2i = d2 if and only if s = 1 and d = n.

We have shown that (1), (2), and (5) are equivalent. Because the quotient R/CH(D)commutes with arbitrary base changes (this is Lemma 1.1.8.6), and a k-algebra R is centralsimple of rank d2 if and only if R ⊗k k is a rank d2 matrix algebra, we see that (4) ⇐⇒(2). Now recall from Lemma 1.2.3.1 that the kernel of the natural surjection

R/CH(D) −→ R/ ker(D)

is nilpotent and equal to the Jacobson radical of R/CH(D). Clearly (4) implies (3), sincethe kernel ker(D) is non-trivial by Lemma 1.1.6.6(2). To complete the proof, we show that(3) implies (5). Since the kernel is stable under separable extensions by Lemma 1.2.3.5 andthe central simple algebra R/ ker(D) of finite rank is split by a finite separable extensionk′/k, we have that Md(k

′) ∼= R/ ker(D) ⊗k k′ ∼= (R ⊗k k′)/ ker(D ⊗k k′). We can choosex1, . . . , xd2 in R to be lifts of a k-basis for R/ ker(D); as this k-basis is also a k′-basis for(R⊗k k′)/ ker(D⊗k k′) ∼= Md(k

′) and (D⊗k k′)(t− xi⊗ 1) = D(t− xi), (5) follows from thenondegeneracy of the trace pairing on Md(k

′).

We can derive from these equivalences the fact that the locus of absolutely irreduciblepseudorepresentations is open. First we give a definition.

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Definition 1.3.4.2. We write PsIrrdR ⊂ PsRdR for the subfunctor of PsRd

R cut out by thefollowing condition: for B ∈ AlgA and D ∈ PsRd

R(B), we say that D ∈ PsIrrdR provided thatfor every B-field k, D ⊗B k : R⊗B k → k is an absolutely irreducible pseudorepresentation.

Corollary 1.3.4.3 (cf. [Che11, Example 2.20]). The subfunctor PsIrrdR ⊂ PsRdR is

Zariski open and therefore representable.

Proof. We use condition (5) of Definition/Lemma 1.3.4.1: choose r1, . . . , rd2 such that(5) holds. This defines a morphism of affine SpecA-schemes

PsRdR −→Md2

D 7→ (Λ1(rirj))i,j,

and Definition/Lemma 1.3.4.1 tells us that the absolutely irreducible locus is the inverseimage of the open subscheme GLd2 ⊂Md2 , which is therefore an open subscheme.

As we will discuss in §2.1.3, the deformation theory of absolutely irreducible pseudorep-resentations is especially nice. It amounts to deforming the absolutely irreducible represen-tation associated to it by Theorem 1.3.1.1; this is already suggested by Corollary 1.3.4.3

The next most tractable case for the deformation theory of pseudorepresentations (whichwe will discuss in §2.1) is the multiplicity free case, which we now define. While “multiplicityfree” is defined over any field k by using the base change to the algebraic closure, just like thecase for “absolutely irreducible,” we will sometimes require that the pseudorepresentationbe realizable as the determinant of a matrix algebra-valued representation over k. We definethe term split for this purpose.

Definition/Lemma 1.3.4.4 ([Che11, Definition 2.19]). Given a d-dimensional pseu-dorepresentation D : R → k, we say that D : R → k is multiplicity free provided thatD ⊗k k is the determinant of a direct sum of pairwise non-isomorphic irreducible k-linearrepresentations. In the notation of Theorem 1.3.1.3, it is equivalent to say that mi = qi = 1for each i.

Call D split provided that it is induced by the determinant of a representation R →Md(k). Equivalently, D is split if and only if R/ ker(D) is a finite product of matrix algebrasover k.

Proof. We will prove the equivalence of the definitions of “split.” If R/ ker(D) is afinite product of matrix algebras

∏s1Mni(k), then by Proposition 1.1.11.5(2) and Proposition

1.1.7.10, D is a product of powers of the determinants of each Mni(k), say D = ⊕ detmiMni,

where∑s

1 nimi = d. If Mi is the representation of R corresponding to R→Mni(k), then wecan recover D as the determinant of the d-dimensional representation ⊕M⊕mi

i .Conversely, assume that R/ ker(D) is not a finite product of matrix algebras. In this

case, R/ ker(D) is nonetheless semisimple with additional properties prescribed by Theorem1.3.1.3: it is a product of simple k-algebras Si, each of which is of finite dimension n2

i overits center ki, where ki/k has exponent (fi, qi). The k-valued pseudorepresentations of Si aredescribed in Lemma 1.3.2.3. Using Proposition 1.1.11.5(2) and Lemma 1.3.2.3 in the sameway as above, D = ⊕ detmiSi for some non-negative integers mi, and d =

∑s1 fiqimini. We

note that ki has separable degree fi over k, and inseparable degree at least qi over k. Wenote that any representation of Si has dimension at least fiqini over k, and this is achievedif and only if Si is a matrix algebra over ki. Since at least one Si is not a matrix algebra by

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assumption, we see that D cannot possibly be realized as the determinant of a d-dimensionalsum of representations of the Si.

Write RepR(k) for the abelian category of finite-dimensional representations of the k-algebra R over k. To be precise, an object of this category is a finite-dimensional k-vectorspace V with a k-linear action of R. We give the following definitions in the context ofrepresentations of algebras; the second term comes from the theory of quiver representations.

Definition 1.3.4.5. Let C be an abelian category.

(1) The Grothendieck group of C, denoted K0(C), is the quotient of the free abeliangroup on the objects of C by the subgroup generated by exact sequences, i.e. by[M ′]− [M ] + [M ′′] where

0 −→M ′ −→M −→M ′′ −→ 0

is an exact sequence in C.(2) Assuming that any object of C has a unique composition series, the Grothendieck

semi-group is the set of isomorphism classes of semisimple objects of C, with theoperation coming from the direct sum of objects.

(3) The dimension vector of an object of C is its image in the Grothendieck groupK0(C).

(4) If any element ρ of C has a composition series, we consider the dimension vector βρto be a vector with respect to the basis of K0(C) given by simple objects.

In the case that C is RepR(k), the finite-dimensional restriction shows that any objecthas a composition series (the representation factors through a subalgebra of Endk(V ); applythe Hopkins-Levitsky theorem). From this, we deduce that K0(RepR(k)) is generated by thesimple finite-dimensional representations of R over k. We can think of the dimension vectorof a representation (or its semisimplification) as a vector with respect to this basis.

Using this basis for K0(RepR(k)), one can say that a pseudorepresentation D : R → kis absolutely irreducible when the associated element of K0(RepR(k)) has a single non-zero entry, which is 1. The pseudopresentation is multiplicity free when the correspondingrepresentation has dimension vector with coordinates consisting of 0 and 1.

1.4. Moduli Spaces of Representations

Let S be an affine Noetherian scheme and let R be a finitely generated, not necessarilycommutative quasi-coherent OS-algebra, which amounts to a finitely generated Γ(OSpecS)-algebra. We consider moduli spaces of representations of R over S-schemes. The Noetherianhypothesis on S will allow for the moduli spaces of representations of R that we will de-scribe below to be Noetherian as well (also cf. Remark 1.1.10.16). We will conclude thissection by drawing a morphism from these moduli spaces of representations to their inducedpseudorepresentation.

1.4.1. Moduli Schemes and Algebraic Stacks. The following definitions describethe functors and groupoids of representations of R that we will study.

Definition 1.4.1.1. With S and R as above and a positive integer d, define the followingS-functors and S-groupoids of d-dimensional representations over an S-scheme X.

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(1) Define the functor on S-schemes Rep,dR by

X 7→ OX-algebra homomorphisms R⊗OS OX −→Md(X).

(2) Define the S-groupoid RepdR by

ob RepdR(X) = V/X rank d vector bundle,

OX-algebra homomorphism R⊗OS OX −→ EndOX (V )

(3) Define the S-groupoid Repd

R by

ob Repd

R(X) = E a rank d2OX-Azumaya algebra,

OX-algebra homomorphism R⊗OS OX −→ E

The functor Rep,dR is of natural interest, but we will often be interested in studying rep-

resentations of R up to isomorphism, where isomorphisms come from conjugation. Explicitly,we say that ρ, ρ′ ∈ Rep,d

R (SpecA) are equivalent when there exists some g ∈ GLd(A) such

that ρ = g−1 · ρ′ · g. We fix this adjoint action of GLd or PGLd on Rep,dR , and we desire a

scheme that represents the functor of orbits of this action.However, the functor sending SpecA to the set of such equivalence classes/orbits – we

could say that it is the functor sending an S-scheme X to a free module with an actionof R, up to isomorphism – is not representable in general. Projective modules must beallowed in order to put equivalence classes of representations of R into families and stillretain representability by an algebraic object. There are two possible strategies that havebeen explored most. One strategy is to find the S-scheme which does the best possible job,by some standard, in representing the moduli problem up to isomorphism. This approachof “geometric invariant theory” will be discussed in the next section §1.5. Here, we willfollow the other approach, which is to remember the data of the isomorphisms betweenobjects, resulting in groupoids fibered over the category of S-schemes that are representableby algebraic stacks. As we will see below (Theorem 1.4.1.4), the groupoids described abovewill naturally arise as the quotient stacks of the adjoint action.

There is a canonical 1-equivalence to the functor (better, S-setoid) Rep,dR from the S-

groupoid whose fiber over an S-scheme X is the data of a free, rank d OX-module, a basis,and an OX-linear action of R⊗OSOX . Having drawn this equivalence, we observe that thereare canonical maps

(1.4.1.2) Rep,dR −→ RepdR −→ Rep

d

R

where the first arrow is given by forgetting the basis and retaining the free rank d vectorbundle with its action, and the second arrow is given by forgetting the vector bundle andretaining the homomorphism from R ⊗OS OX into its bundle of endomorphisms. We note

that the Azumaya algebras in Repd

R are not taken up to equivalence8, so that they corespondup to isomorphism with PGLd-torsors, not elements of the Brauer group. In other words, weconsider non-trivial but locally isotrivial (i.e. Zariski locally trivializable) Azumaya algebras.

Theorem 1.4.1.3. Let S,R, and d be as above. Then the functor Rep,dR is representable

by an affine finite type S-scheme.

8Azumaya algebras E1, E2 over A are called equivalent if there exist finite rank projective modules V1, V2 suchthat there exists an isomorphism of A modules E1 ⊗A EndA(V1) ∼= E2 ⊗A EndA(V2). The Azumaya-Brauergroup classifies Azumaya A-algebras up to equivalence, cf. [Gro68, §2].

62

Proof. Choose a set of generators r1, . . . , rn for R over A. Then we have a morphismof functors

Rep,dR −→Mn

d

ρ 7→ (ρ(r1), ρ(r2), . . . , ρ(rn)).

Let X be the finite set X = x1, . . . , xn and let F be the non-commutative quasi-coherentS-algebra freely generated by X. We observe that the map above induces an isomorphismRep,d

F

∼→ Mnd . There is a canonical map F X given by sending xi 7→ ri for each

i, 1 ≤ i ≤ n, and let J ⊂ F be its kernel, which is a two-sided ideal of F . For f ∈ J ,consider it as a function f(x1, . . . , xn) of the free variables xi. There exists a morphismW f ∈ HomS−schemes(M

nd ,Md) corresponding to f , given by sending an n-tuples of d × d-

dimensional matrices (m1, . . . ,mn) to f(m1, . . . ,mn). Let W fij ∈ Γ(O(Mn

d )) be the regular

function obtained from composing W f with the projection onto the (i, j)th coordinate of

d × d-matrices, and let IJ be the ideal of Γ(O(Mnd )) generated by W f

ij as f varies overelements of J and 1 ≤ i, j ≤ d.

We claim that the closed subscheme V (IJ) ⊂Mnd represents the functor Rep,d

R is isomor-

phic to Rep,dR under the map above. Clearly we have a monomorphism Rep,d

R → V (IJ) ⊂Mn

d , because each of the relations f ∈ J are sent to zero under the representation. For

any affine S-scheme SpecA, the map of sets Rep,dR (A) → V (IJ)(A) is surjective, since for

(m1, . . . ,mn) ∈ V (IJ)(A), the A-algebra homomorphism R ⊗Γ(OS) A→ Md(A) arising fromsending ri to mi defines a representation which maps to (m1, . . . ,mn).

We recall that GLd and PGLd act on Rep,dR via the adjoint action, conjugating the

matrix coefficients of the representations.

Theorem 1.4.1.4. The groupoids RepdR and Repd

R are equivalent to algebraic stacks, inparticular the quotient algebraic stacks

RepdR∼= [Rep,d

R /GLd], Repd

R∼= [Rep,d

R /PGLd].

The canonical smooth presentation maps of these quotient stacks

Rep,dR −→ [Rep,d/GLd] −→ [Rep,d

R /PGLd]

correspond to the natural maps of groupoids (1.4.1.2).

For the reader’s convenience, we recall some equivalent definitions of Azumaya algebras.

Definition 1.4.1.5 ([Gro68, Theorem 5.1]). Let X be a scheme, and let E be a coherentOX-module which has the structure of a OX-algebra. Then we say that E is an Azumayaalgebra if one of the following equivalent conditions are satisfied.

(1) E is locally free as a OX-module, and for every x ∈ X, the fiber E ⊗OX κ(x) is acentral simple algebra.

(2) E is locally free as a OX-module, and the canonical homomorphism E ⊗OX Eop →EndOX (E) is an isomorphism.

(3) There exists an etale covering U → X such that E ⊗OX OU ∼= Md(OU) for somed ≥ 1.

Now we prove Theorem 1.4.1.4.

Proof. The quotient [Rep,dR /GLd] parameterizes, by definition, a GLd-torsor G along

with a GLd-equivariant map G → Rep,dR . This is what we will create from the data of an

63

SpecA-point of RepdR, i.e. the data (V/A, ρ : R ⊗OS A → EndA(V )), where V is a rankd projective A-module. We create a GLd torsor over SpecA corresponding to V , settingits functor of points on SpecA-schemes X to be G(X) := IsomOX (V ⊗A OX ,O⊕dX ). Thisdefines an equivalence of categories between GLd-torsors up to isomorphism and rank dlocally free sheaves up to isomorphism. The identity map in G(G) = IsomOG(V ⊗AOG,O⊕dG )

is a canonical isomorphism V ⊗A OG∼→ O⊕dG . This defines a OG-linear action of R ⊗OS OG

on the free vector bundle O⊕dG with its canonical basis, so that we have a map G → Rep,dR .

It remains to show that this map is GLd-equivariant. The action of GLd on G on the rightcomes from the standard action of GLd(X) on O⊕dX . This is effectively the basis changeaction of GLd on the map R ⊗OS OX → Md(X), which is the adjoint action. This is an

A-point of [Rep,dR /GLd], as desired.

For the inverse construction, we take an A-point of [Rep,dR /GLd], i.e. a GLd-equivariant

map G → Rep,dR , and create an object of RepdR(A). We use the equivalence of categories

between vector bundles and GL-torsors mentioned above to find a rank d projective A-module such that G(X) ∼= IsomOX (V ⊗A OX ,O⊕dX ) for all A-schemes X. As V ⊗A OG is arank d-free module with a canonical basis as discussed above, we can take our initial dataof R⊗OS OG →Md(G) and compose it with the canonical map Md(G)

∼→ EndOG(V ⊗A OG),to obtain an action of R ⊗OS OG on V ⊗A OG. We leave it as an exercise to show that the

GLd-equivariance of G → Rep,dR is then exactly what we need in order to descend this map

to SpecA.

The proof that Repd

R∼= [Rep,d

R /PGLd] goes along the same lines. We choose a SpecA-

point of Repd

R: a map R⊗OS A→ E, where E is a rank d2 Azumaya A-algebra. We can thencreate a PGLd-torsor G whose X-points for an A-scheme X are G(X) := IsomOX−alg(E ⊗AOX ,Md(X)), and the action of PGLd(X) on G(X) comes from its adjoint action on Md(X).

Then the identity map id ∈ G(G) corresponds to a canonical isomorphism E⊗AOG∼→Md(G)

defining a morphism G → Rep,dR , and we observe that the adjoint action on both the source

and target make this map PGLd-equivariant, and therefore an A-point of [Rep,dR /PGLd].

For the inverse construction, from an A-point G → Rep,dR of [Rep,d

R /PGLd] we constructa rank d2 Azumaya A-algebra E so that there is a canonical isomorphism of coherent OG-algebras E ⊗A OG

∼→ Md ⊗OG. Then the map R ⊗OS OG → Md(G) can be composed with

E(G)∼→Md(G) to get a map R⊗OS OG → E⊗AOG. The PGLd-equivariance of G → Rep,d

R

allows us to descend this map from G to SpecA.The claim that the forgetful maps from Rep,d

R and the presentation maps commute withthe equivalences we have drawn follows from checking that the universal framed representa-

tion over Rep,dR induces a map to RepdR (resp. Rep

d

R) compatible with the universal objecton the quotient stack via the correspondence that we have written out above.

1.4.2. Mapping Algebraic Stacks of Representations to the Moduli Schemeof Pseudorepresentations. Let X be an S-scheme. Having defined these moduli spacesof representations of the OS-algebra R, we know that the association of an X-valued repre-sentation of R, that is, the data

(ρ : R⊗OS OX −→Md(OX)) ∈ Rep,dR (X)

64

to an X-valued pseudorepresentation by taking the determinant (see Theorem 1.1.7.4(6) andRemark 1.1.7.6)

R⊗OS OXρ−→Md(OX)

det−→ OXdefines a morphism of S-schemes

(1.4.2.1) ψ : Rep,dR −→ PsRd

R.

Think “ψ” for pseudorepresentation.We will now show that there is also a natural pseudorepresentation associated to objects

of RepdR(X) and Repd

R(X) that is constant across isomorphism classes in the groupoid, sothat there are morphisms of algebraic stacks

ψ : RepdR → PsRdR, ψ : Rep

d

R → PsRdR

which commute with the canonical maps (1.4.1.2). Then we will have a commutative diagram

(1.4.2.2) Rep,dR

(1.4.1.2)//

ψ

))

RepdR(1.4.1.2)

//

ψ

##

Repd

R

ψ

PsRdR

All that we need to do is construct the vertical arrow ψ, sending, for X an S-scheme, anAzumaya OX-algebra-valued representation R ⊗OS OX → E to an OX-valued pseudorepre-sentation. We will achieve this using the reduced norm map out of any Azumaya algebra,and indeed, the rest of the characteristic polynomial coefficients. We construct these coeffi-cient functions as follows. Each coefficient of the characteristic polynomial defines a regularfunction Md → A1 which is invariant under the adjoint action of PGLd. Each Azumayaalgebra E is a form of Md twisted by this action (cf. [Gro68, Corollary 5.11]); therefore, thecharacteristic polynomial function descends from E⊗OXOU ∼= Md(OU) to E over OX [Gro68,5.13].

Now there are at least two perspectives we could take on the pseudorepresentation asso-

ciated to an object ρ : R⊗OSOS → E of Repd

R(X). We can compose this representation withthe reduced norm, which we continue to write as “det” as it is equal to det etale-locally:

R⊗OS OSρ−→ E det−→ OX

is compatible with base change, making a pseudorepresentation. Alternatively, as PsRdR is a

scheme, it is a sheaf on the etale site SEt, so that we can choose an etale cover U of X anddescend the pseudorepresentation

R⊗OS OUρ⊗OU−→ E ⊗OX OU ∼= Md(OU)

det−→ OUto a pseudorepresentation over OX .

In any case, we have completed the construction of the diagram (1.4.2.2).

1.4.3. Representations Factor through the Universal Cayley-Hamilton Alge-bra. In this paragraph, we will show that our basic assumptions – that R is a finitely gen-erated algebra over an affine Noetherian base – are sufficient to show that the d-dimensionaluniversal representation of R factors through an algebra finite over its center. This is a con-sequence of the theorem below, which shows that any representation of R factors through

65

the universal d-dimensional Cayley-Hamilton representation associated to R,

ρu : R⊗A ΓdA(R)ab −→ E(R, d)

Recall the definition of Cayley-Hamilton representations from §1.2.4. In particular, E(R, d)is a ΓdA(R)ab-algebra, defined to be (R⊗A ΓdA(R)ab)/CH(Du).

Theorem 1.4.3.1. Any representation in Rep,dR (B) (resp. RepdR(B), resp. Rep

d

R(B)) ofR factors uniquely through the universal Cayley-Hamilton representation

ρu ⊗ΓdA(R)ab B : R⊗A B −→ E(R, d)⊗ΓdA(R)ab B.

This factorization induces canonical equivalences of PsRdR-schemes (resp. algebraic stacks)

Rep,dR

∼−→ Rep,dE(R,d),Du|E ,

RepdR∼−→ RepdE(R,d),Du|E ,

Repd

R∼−→ Rep

d

E(R,d),Du|E ,

where the left hand side algebraic stacks are considered to be PsRdR-stacks through the map

ψ (resp. ψ, resp. ψ).

The implicit map ΓdA(R)ab → B arises from the determinant (or reduced norm) of therepresentation, along with the representability result Theorem 1.1.7.4.

Remark 1.4.3.2. While the theorem has an especially nice consequence when A is as-sumed to be Noetherian and R is assumed to be finite generated over A, the theorem is truewith or without these finiteness assumptions.

Proof. Let SpecB be an affine PsRdR-scheme (and therefore naturally an A-algebra) and

write ζ : SpecB → PsRdR for the structure map, i.e. a choice of a B-valued d-dimensional

pseudorepresentation of R. Any object of the SpecA-groupoids Rep,dR (B), RepdR(B) induces

an Azumaya B-algebra-valued representation ρ : R ⊗A B → E ∈ Repd

R(B) by the forgetfulmaps (1.4.1.2). The question of the factorization of a representation does not depend on the

forgotten data, so it will suffice to prove the result for ρ. So we choose ρ ∈ Repd

R(B), suchthat

SpecBρ //

ζ ##

Repd

R

ψ

PsRdR

commutes.Recall Definition 1.2.4.1, which is the notion of a Cayley-Hamilton representation of R.

Following Remark 1.2.4.2, we note that a the data of ρ induces a d-dimensional Cayley-Hamilton representation of R over B, namely

(B, (E , det), ρ),

where det : E → B represents the reduced norm map for the Azumaya B-algebra E .Proposition 1.2.4.3 shows that the universal d-dimensional Cayley-Hamilton represen-

tation (ΓdA(R)ab, (E(R, d), Du|E), ρu) is initial in the category CHd(R) of Cayley-Hamiltonrepresentations of R. Thus there exists a canonical CHd(R)-morphism

(ΓdA(R)ab, (E(R, d), Du|E), ρu) −→ (B, (E , det), ρ).

66

This includes the datum of a A-morphism ΓdA(R)ab → B, corresponding to the pseudorep-resentation det ρ by representability and contravariantly equivalent to ζ. There is also acanonical morphism

E(R, d)⊗ΓdA(R)ab B → E ∈ Repd

E(R,d),Du|E(B),

factoring ρ through the canonical quotient map ρu : R⊗A ΓdA(R)ab → E(R, d).

We have therefore exhibited a PsRdR-morphism Rep

d

R → Repd

E(R,d),Du|E .We can derive a quasi-inverse from ρu. Define

η : E(R, d)⊗ΓdA(R)ab B → E ∈ RepE(R,d),Du|E(B).

We get from η a representation of R, η (ρu ⊗B) ∈ Repd

R(B).

When A and R satisfy appropriate finiteness conditions, we know that the universalCayley-Hamilton algebra, the ΓdA(R)ab-algebra E(R, d), is finite as a ΓdA(R) and is a Noether-ian ring. Therefore we may show that the representation theory of a finitely generated algebraover a commutative Noetherian ring reduces to the theory of Noetherian (non-commutative)rings that are finite over their Noetherian center.

Corollary 1.4.3.3. Fix a positive integer d. If A is Noetherian and R is finitely gener-ated as an A-algebra, all of the d-dimensional representations of R factor canonically throughan algebra which is finite as a module over its center and Noetherian, namely, each repre-sentation factors uniquely through

ρu : R⊗A ΓdA(R)ab −→ E(R, d).

Proof. This follows directly from Theorem 1.4.3.1 along with Corollary 1.2.2.10.We recapitulate Theorem 1.4.3.1 for clarity. A B-valued d-dimensional representation

of R amounts to some map ρ : R ⊗A B → E where E is a rank d2 B-Azumaya algebra,possibly with some extra data that we can discard. The induced pseudorepresentation ψ(ρ)induces a map ΓdA(R)ab → B by the representability of PsRd

R. This gives us the B-valuedrepresentation of R⊗AΓdA(R)ab. Then Theorem 1.4.3.1 shows that this representation factorsthrough ρu ⊗ΓdA(R)ab B.

The rest of the statements follow directly from Corollary 1.2.2.10. Since E(R, d) isan ΓdA(R)ab-algebra, the center of E(R, d) contains the image of ΓdA(R)ab in E(R, d), andE(R, d) is finite as a ΓdA(R)ab-module, it must also be module-finite over its center. Asnoted in Corollary 1.2.2.10, these facts along with the Noetherianness of A imply that R isNoetherian as well.

Remark 1.4.3.4. We could prove a version of Corollary 1.4.3.3 with A being a field anddemanding that a base pseudorepresentation D : R → A be fixed. Then the functor of allrepresentations lying over this pseudorepresentation via ψ would factor through the Cayley-Hamilton quotient R/CH(D) of R relative to D, and Theorem 1.3.3.2 gives conditions forthis quotient to be finite dimensional. We will use these ideas later, extending Corollary1.4.3.3 to the case that R is a profinite algebra satisfying an appropriate finiteness condition(see Theorem 3.2.3.2).

1.4.4. Representations of Groups into Affine Group Schemes. In this paragraphwe restrict our attention to representations of group algebras as opposed to general asso-ciative algebras, and then generalize this case to representations of a group valued in anarbitrary group scheme.

67

Let Γ be a finitely generated group. Then R = OS[Γ] is a finitely generated quasi-coherent OS-algebra, and the formalism of the above can be repeated. We leave the readerto verify the following basic equivalences, which amount to saying for a ring A that A-valued d-dimensional representations ρ : Γ → GLd(A) are equivalent to homomorphismsA[Γ]→Md(A).

Proposition 1.4.4.1. Let R = OS[Γ] and let X represent an S-scheme. Then

(1) Rep,dR is naturally equivalent to the functor

X 7→ Γ −→ GLd(X).(2) RepdR is equivalent to the S-groupoid with objects over X being

V/X a rank d vector bundle, Γ −→ AutOX (V )(X).and morphisms being isomorphisms of these data.

(3) Repd

R is naturally equivalent to the S-groupoid with objects over X being

H/X an inner form of GLd,Γ −→ H(X).and morphisms being isomorphisms of these data.

Proof. Omitted.

We also might be interested in representations of Γ that fix certain tensors, for example,representations valued in Spd or SOd. We will simply let G be an arbitrary finite type flataffine S-group scheme and consider the moduli of representations of Γ into G.

Definition 1.4.4.2. For an abstract group Γ and a finite type flat affine S-group schemeG, we define the following functors and S-groupoids.

(1) Let Rep,GΓ denote the functor on S-schemes X

X 7→ homomorphisms Γ −→ G(X).

(2) Define the S-groupoid RepGd by

ob RepGΓ (X) = G a right G-torsor over X,

Γ −→ AutGX(G)(X).

Here AutGX(G) is the X-group scheme of automorphisms of G, where an automor-phism of G over an X-scheme Y is an endomorphism of the Y -scheme G×X Y whichis equivariant for the right action of G×S Y .

(3) Define the S-groupoid RepG

d by

ob RepG

Γ (X) = an inner form H of G over X,

Γ −→ H(X).

We observe that there are natural maps

(1.4.4.3) Rep,GΓ −→ RepGΓ −→ Rep

G

Γ .

To construct the first map, choose a trivial G-torsor G over S so that

(1.4.4.4) AutGS (G)∼−→ G,

where the isomorphism follows from the fact that the maps G → G which are equivariantfor the right action of G on itself are precisely the left translations of G on itself. Then the

68

first map is given by sending (ρ : Γ→ G(X)) ∈ Rep,GΓ (X) to the composition

Γ −→ G(X)∼−→ AutGS (G)(X).

The second map is given by forgetting the G-torsor G over X inducing the inner formH := AutGX(G)(X), where we see that this is an inner form by the isomorphism, for a trivialG-torsor.

Theorem 1.4.4.5. Let Γ, G be as above. Then the functor Rep,Gd is representable by an

affine finite type S-scheme.

Proof. Choose a set of generators γ1, . . . , γn for Γ. Then we have a morphism of functors

Rep,GΓ −→ Gn

ρ 7→ (ρ(r1), ρ(r2), . . . , ρ(rn)).

Any word w on the n letters γ1, . . . , γn induces a map

fw : Gn −→ G

(g1, . . . , gn) 7→ w(g1, . . . , gn).

given by substituting gi for γi. We observe that Gn represents RepGFn , where Fn is thefree group on n letters. A representation of Fn valued in A corresponding to a morphismp : SpecA → Gn induces a representation of Γ if and only if, for every word w in theletters (γi) such that w = id ∈ Γ, fw p ∼= idG ×S SpecA, where idG is the identity section

idG : S → G of the S-group scheme G. Therefore Rep,GΓ is precisely the intersection over

words w such that w = id ∈ Γ of the closed subschemes Gw of Gn given by the fiber product

Gw //

Gn

fw

SidG // G.

As S is Noetherian and G is finite type over S, so is Gw Noetherian and finite type overS.

Just as GLd (or PGLd) acts on group representations Γ → GLd via the adjoint action,so does the adjoint group of G, namely G/Z(G), act on itself by the adjoint action. This

gives an action of G and PG := G/Z(G) on Rep,GΓ . Also, like before, this is a natural

notion of equivalence for the points of Rep,GΓ , but the functor of equivalence classes is not

representable. The following quotient stacks retain the equivariant geometry of Rep,GΓ , and

are equivalent to the stacks of representations defined above.

Theorem 1.4.4.6. The groupoids RepGΓ and RepG

Γ are equivalent to algebraic stacks, inparticular the quotient algebraic stacks

RepGΓ∼= [Rep,G

Γ /G], RepG

Γ∼= [Rep,G

Γ /PG].

The canonical flat presentation maps of these quotient stacks

Rep,GΓ −→ [Rep,G/G] −→ [Rep,G

Γ /PG]

correspond to the natural maps of groupoids (1.4.4.3).

Proof. Let G1 be a right G-torsor over an S-scheme X, equipped with a group homo-morphism Γ → AutGX(G1)(X), where we use the superscript to denote various copies of the

69

same G-torsor. We wish to induce from this data a G-equivariant map G2 → Rep,GΓ . We

know that G2 → X trivializes G1 via the map

(1.4.4.7)G×X G2 ∼−→ G1 ×X G2

(g, x) 7→ (xg, x),

so we have a map

(1.4.4.8) Γ −→ AutGG2(G1 ×X G2)(G2)∼−→ GG2(G2)

inducing a map G2 → Rep,GΓ . Now we wish to show that (1.4.4.8) is G-equivariant for the

standard right action of G on the left and the adjoint action of G on the right.The right action of g′ ∈ G on G2 on the right side of (1.4.4.7) sends

(xg, x) 7→ (xg, xg′) = (xg′g′−1g, xg′),

and therefore acts on the left side of (1.4.4.7) by

(g, x) 7→ (g′−1g, xg′).

so its action on GG2 is the right action by multiplication on the left by the inverse. Now weneed to consider GG2 as a trivial right G-torsor and calculate the induced intertwining actionon the functor of automorphisms of GG2 as a torsor. These automorphisms are precisely theleft translations by g′′ ∈ G, g 7→ g′′g. The intertwining action of g′ ∈ G on this map is then

(g, x) 7→ (g′g, xg′−1) 7→ (g′′g′g, xg′−1) 7→ (g′−1g′′g′−1g, x),

which is the adjoint action, as desired.For the inverse construction, we start with a G-equivariant map from a G-torsor G2 over

an S-scheme X with a G-equivariant map G2 → Rep,GΓ . By definition of Rep,G

Γ , thereexists a homomorphism

Γ −→ G(G2).

As G2/X is trivialized by G1 → X, we can fix an isomorphism AutGG1(G2 ×X G1) ∼= GG1 , andreplace G with this expression in the homomorphism above. We leave it as an exercise tocheck that the G-equivariance of G2 → Rep,G

Γ is exactly what we need in order to descendthe automorphisms of G2 ×X G1 from G1 to X.

1.5. Geometric Invariant Theory of Representations

In the previous section, we defined the affine, finite type S-scheme Rep,dR of d-dimensional

representations of the quasi-coherent, finitely generated OS-algebra R. After making note ofthe natural equivalence relation of conjugation, we defined the algebraic quotient S-stacksarising from this action. These algebraic stacks have a clear, explicit description as an S-groupoid. In this section, we will study an alternative approach, using geometric invarianttheory to find the “best possible” S-scheme to stand in for a quotient of Rep,d

R . Geometricinvariant theory (GIT) was originally developed by Mumford (see e.g. [Mum65]). We will firstdescribe Alper’s theory of adequate moduli spaces [Alp10], which summarizes and generalizesthe results of geometric invariant theory in a way that will be useful for our purposes, asdescribes nicely the relationship between the quotient stack and the GIT quotient schemevia the canonical projection morphism.

1.5.1. Alper’s Theory of Adequate Moduli Spaces. Say that an affine algebraicgroup G acts on a finite type affine scheme X = SpecA over a field k. The GIT quotient

70

scheme, which we will denote X//G, is the spectrum of the invariant regular functions onX. That is, X//G := SpecAG, where AG ⊂ A is the k-subalgebra of A of co-invariants ofthe co-action A → O[G] ⊗k A. We have a natural map X → X//G. When G is reductive,Mumford’s theory implies that X//G is finite type over Spec k and the map X → X//G hasappropriate universal properties of the quotient of a group action. The finite type propertyof the quotient is not necessarily true when G is not reductive [Nag60]. Now we turn ourinterest toward the relationship of the quotient stack [X/G] to the GIT quotient schemethrough the canonical morphism φ : [X/G]→ X//G. We write X := [X/G] for short.

As Alper notes [Alp10, p. 2], φ can be checked to have the special properties

(1) For any surjection of quasi-coherent OX -algebras A → B and section t ∈ Γ(X ,B),there exists an integer N > 0 and a section s ∈ Γ(X ,A) such that s 7→ tN .

(2) AG → Γ(OX ) is an isomorphism.

Slight extension of these properties to apply locally on non-affine spaces give the definitionalconditions for φ to be an adequate moduli space. As Alper puts it, “it turns out thatproperties (1) and (2) capture the stack-intrinsic properties of such GIT quotient stacks[X/G] and that these properties alone suffice to show that the quotient X//G inherits nicegeometric properties” [Alp10, p. 2].

Definition 1.5.1.1 ([Alp10]). A quasi-compact and quasi-separated morphism φ : X →Y from an algebraic stack to an algebraic space is an adequate moduli space if the followingtwo properties are satisfied:

(1) For every surjection of quasi-coherentOX -algebrasA B and every etale morphismp : U = SpecA → Y and section t ∈ Γ(U, p∗φ∗B) there exists N > 0 and a sections ∈ Γ(U, p∗φ∗A) such that s 7→ tN , and

(2) OY → φ∗OX is an isomorphism.

The first property is called “adequately affine,” and indeed, any quasi-compact, quasi-separated map of algebraic spaces that is adequately affine is affine [Alp10, Theorem 4.3.1],generalizing Serre’s criterion for affineness (which is the same condition with N = 1). Insum, we require the following notions of adequacy.

Definition/Lemma 1.5.1.2. Let A→ B be a homomorphism of rings. Let X → Y bea morphism of algebraic spaces.

(1) We call A → B adequate if for all b ∈ B, there exists some N > 0 and a ∈ A suchthat a 7→ bN .

(2) We call A → B universally adequate if for all A-algebras A′, A′ → A′ ⊗A B isadequate.

(3) We call X → Y an adequate homeomorphism if its is an integral, universal homeo-morphism which is a local isomorphism at all points with a residue field of charac-teristic zero. In particular, SpecB → SpecA is an adequate homeomorphism if andonly if(a) ker(A→ B) is locally nilpotent (i.e. every element is nilpotent),(b) ker(A→ B)⊗Q = 0, and(c) A→ B is universally adequate.

Proof. The “if and only if” statement is [Alp10, Proposition 3.3.5(2)].

We will be interested in adequate moduli spaces that arise from the conventional GITsetting, where a reductive group scheme acts on a scheme.

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Example 1.5.1.3 ([Alp10, Theorem 9.1.4]). Let S be an affine scheme and letX = SpecAbe an affine S-scheme. Let G be a reductive group S-scheme with an action on X. Then

φ : [X/G] −→ SpecAG

is an adequate moduli space.

Here is Alper’s main theorem on adequate moduli spaces.

Theorem 1.5.1.4 ([Alp10, Main Theorem]). Let φ : X → Y be an adequate moduli space.Then

(1) φ is surjective, universally closed, and universally submersive.(2) Two geometric points x1, x2 ∈ X (k) are identified in Y if and only if their closures

x1 and x2 in X ×Z k intersect.(3) If Y ′ → Y is any morphism of algebraic spaces, then X ×Y Y ′ → Y ′ factors as

an adequate moduli spaces X ×Y Y ′ → Y followed by an adequate homeomorphism

Y → Y ′.(4) Suppose X is finite type over a Noetherian scheme S. Then Y is finite type over S

and for every coherent OX -module F , φ∗F is coherent.(5) φ is universal for maps from X to algebraic spaces which are either locally separated

or Zariski-locally have affine diagonal.

Remark 1.5.1.5. We note that adequate moduli spaces φ : X → Y share particularsimilarities with both affine morphisms of schemes and proper morphisms of schemes. Indeed,an adequate moduli space is adequately affine (part (1) of Definition 1.5.1.1), and as we notedabove, a quasi-compact, quasi-separated morphism of algebraic spaces is adequately affine ifand only if it is affine. On the other hand, φ is universally closed and φ∗ preserves coherentsheaves, which are characteristics of proper morphisms. Since a morphism of schemes that isboth affine and proper is finite, we expect φ to behave somewhat like a finite morphism andmoreover, by part (2) of the Definition 1.5.1.1, like an isomorphism! The obstruction to beingan isomorphism is the lack of representability and the accompanying lack of separatedness(fact: a quotient stack of a separated scheme is separated if and only if all stabilizers arefinite). This “isomorphism up to lack of representability” property is encapsulated moreprecisely in part (5) of the theorem (1.5.1.4).

Remark 1.5.1.6. One important notion from geometric invariant theory that will be usedin the sequel is the following two facts about orbits (of geometric points) of the action of areductive group on a scheme. Working over an algebraically closed field, let a reductive groupG act on an variety X, which for simplicity we assume to be affine. Because X = SpecAis affine, every orbit is semistable. The standard fact from geometric invariant theory isthat every semistable orbit contains a unique closed semistable orbit (one can get this bycombining Example 1.5.1.3 and part (2) of the theorem above). Now, obviously an invariantregular function on X must remain constant along an orbit. Moreover, it must remainconstant along an orbit’s closure, since regular functions are “continuous.” This means thatinvariant regular functions cannot distinguish orbits whose closures overlap! It turns out thatthe geometric points of X//G := SpecAG are in bijective correspondence with the orbits ofG in X modulo the equivalence relation of overlapping closure. This is what part (5) ofTheorem 1.5.1.4 expresses.

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1.5.2. Geometric Invariant Theory on Rep. Example 1.5.1.3 shows that in theclassical setting of geometric invariant theory, where a reductive group G acts on an affinescheme X, the resulting morphism [X/G]→ X//G is an adequate moduli space. By Theorem

1.4.1.4, the algebraic stacks RepdR (resp. Repd

R) are quotient stacks for the adjoint action of

GLd (resp. PGLd) on the finite type affine S-scheme Rep,dR . Therefore, as Rep,d

R is an affine

scheme and Rep,dR //GLd ∼= Rep,d

R //PGLd, each of the morphisms

(1.5.2.1)φ : RepdR −→ Rep,d

R //PGLd,

φ : Repd

R −→ Rep,dR //PGLd

are adequate moduli spaces. Therefore by the universality of the GIT quotient schemefor maps to separated schemes (Theorem 1.5.1.4(5)) we can canonically factor the diagram(1.4.2.2) to get a diagram

(1.5.2.2) Rep,dR

//

ψ

$$

φ

))

RepdR //ψ

φ

##

Repd

R

ψ

φ

GIT

ν

PsRdR

where GIT stands in for the GIT quotient scheme Rep,dR //PGLd. To put these ideas in

words, the maps ψ, ψ, ψ of (1.4.2.2) factor uniquely through the GIT quotient.

Remark 1.5.2.3. One shortcoming of the GIT quotient is that despite the concretemoduli problem that Rep,d

R and the other moduli stacks solve, this does not lend us acomplete description of the GIT quotient in terms of a “functor of points.” Its one universalproperty is that of Theorem 1.5.1.4(5), but this characterizes morphisms out of it insteadof its functor of points. However, we do know the “functor of geometric points” of the GITquotient, following Remark 1.5.1.6: geometric points of a GIT quotient of an affine schemecorrespond to closed orbits of geometric points. In the next paragraph, we will discover whatthese closed orbits in Rep,d

R are in terms of its moduli problem. But we emphasize that

this is a property of Rep,dR //PGLd and not a characterization, since the geometric points

of a scheme do not characterize it. For another, related shortcoming of GIT quotients, seeRemark 1.5.4.4.

As noted in the introduction, one of the main ideas behind pseudorepresentations is toserve as a concrete (i.e. a moduli problem) replacement for the GIT quotient Rep,d

R //PGLd.We will therefore be very interested in the map

(1.5.2.4) ν : Rep,dR //PGLd −→ PsRd

R,

which we expect to be nearly an isomorphism (see Theorem 1.5.4.2).

1.5.3. Work of Kraft, Richardson, et al. on Orbits of the Adjoint Action onRepresentations. In this paragraph we describe the geometric points of the GIT quo-tient scheme Rep,d

R //PGLd of the adjoint action of PGLd on Rep,dR . As we noted in

Remark 1.5.1.6 following Theorem 1.5.1.4(2), these geometric points correspond naturally

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and bijectively to closed orbits in Rep,dR , or, equivalently, closed geometric points in RepdR

(resp. Repd

R). So what are the closed geometric points in RepdR?Kraft [Kra82] answered this question, proving the following

Theorem 1.5.3.1 ([Kra82, §II.4.5, Proposition]). As usual, let R be a finitely generatedquasi-coherent OS-algebra where S is an affine Noetherian scheme. For any algebraicallyclosed S-field k, the following equivalent statements are true.

(1) The closed orbits of PGLd(k) in Rep,dR (k) are precisely the orbits of semisimple

d-dimensional representations of R⊗OS k.

(2) The closed geometric points of RepdR (resp. Repd

R) are in natural bijective corre-spondence with isomorphism classes of semisimple d-dimensional representations ofR⊗OS k.

(3) The geometric points of the GIT quotient affine scheme Rep,dR //PGLd are in natural

bijective correspondence with the semisimple d-dimensional representations of R⊗OSk.

This result uses the Hilbert-Mumford criterion.

Remark 1.5.3.2. This theorem implies that the canonical map ν : Rep,dR //PGLd −→

PsRdR of (1.5.2.4) induces a bijection on geometric points! We will take up this point in

the following paragraph, spending this paragraph on the proof of Theorem 1.5.3.1 and itsanalogue Theorem 1.5.3.7, which addresses Rep,G

Γ in place of Rep,dR .

Proof. Part (1) is due to Kraft [Kra82, §II.4.5, Proposition]. The equivalence of (1)with (2) and (3) follows from Remark 1.5.1.6.

Richardson [Ric88] answered this question in the case that R is a group algebra; in fact,his proof addresses representations of a finitely generated group Γ into a reductive group G(see the setup for these representation moduli schemes/stacks in §1.4.4), with G = GLd asa special case. The techniques of his proof were improved by several people, with notablecontributions (for our purposes) of Serre [Ser05] (following [Ser98, Part II]) and Bate-Martin-Rohrle [BMR05]. These are the results that we now overview. They can be summarized in

brief by saying that the closed orbits of the adjoint action ofG on Rep,GΓ over an algebraically

closed field (or, equivalently, the closed geometric points in RepGΓ or RepG

Γ ) are in naturalbijective correspondence with “semisimple” representations. Of course, we must say whatsemisimple means in G.

We work over an algebraically closed field k.

Definition 1.5.3.3 ([Ser05]). A subgroup H ⊂ G(k) is called G-completely reducibleprovided that whenever H is contained in some parabolic subgroup P of G, it is containedin a Levi subgroup of P .

This generalizes the familiar case from GLd: if H ⊂ GL(V ), then V is a semisimpleH-module if and only if H is GL(V )-completely reducible. By the same token, we give anotion of semisimplicity for a reductive group-valued homomorphism.

Definition 1.5.3.4. Let G be a reductive group over an algebraically closed field k, andlet Γ be a group. We say that a homomorphism ρ : Γ → G(k) is semisimple if ρ(G) isG-completely reducible.

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One would hope that a result analogous to Theorem 1.5.3.1 can be proved with G inplace of GLd. The basic problem is that in positive characteristic, a reductive subgroup of areductive group may not be semisimple. Richardson proved a result to this effect, overcomingproblems in positive characteristic, although it was originally proved with the notion of astrongly reductive subgroup in place of a completely reducible subgroup.

Definition 1.5.3.5 ([Ric88, Definition 16.1]). Let G be a reductive group over an al-gebraically closed field. Let H be a closed subgroup of G and let S be a maximal torus ofCG(H), the centralizer in G of H. We call H a strongly reductive subgroup of G providedthat H is not contained in any proper parabolic subgroup of CG(S).

Richardson’s definition is set up in order to apply geometric invariant theory – in partic-ular, the Hilbert-Mumford numerical criterion – to show that the closed orbits of the adjointaction on G (or PG) on Rep,G

Γ correspond to strongly reductive subgroups. Here, the sub-group in question is the closure of the image of the representation. It was more recentlyproved that strong reductivity is the same as complete reducibility.

Theorem 1.5.3.6 ([BMR05, Theorem 3.1]). Let G be a reductive algebraic group over analgebraically closed field k, and let H be a closed algebraic subgroup. Then H is G-completelyreducible if and only if H is strongly reductive in G.

From this, the desired result follows.

Theorem 1.5.3.7. As usual, let Γ be a finitely generated group and let G be a reductiveS-group scheme. For any algebraically closed S-field k, the following equivalent statementsare true.

(1) The closed orbits of G(k) (resp. PG(k)) in Rep,GΓ (k) are precisely the orbits of

semisimple representations Γ→ G(k).

(2) The closed geometric points of RepGΓ (resp. RepG

Γ ) are in natural bijective correspon-dence with isomorphism classes of semisimple representations Γ→ G(k).

(3) The geometric points of the GIT quotient affine scheme Rep,GΓ //PG are in natural

bijective correspondence with semisimple representations Γ→ G(k).

Proof. Richardson proved that the n-tuples of geometric points of G whose orbit underthe adjoint action of G or PG is closed are precisely those n-tuples whose generated subgroupof G is strongly reductive [Ric88, Theorem 16.4]. This is equivalent to statement (1) byTheorem 1.5.3.6. The equivalence of (1) with (2) and (3) is clearly, in light of Remark1.5.1.6.

1.5.4. The GIT quotient and PsRdR are Almost Isomorphic. In this paragraph,

we will show that the canonical map ν : Rep,dR //PGLd → PsRd

R is a finite universal home-omorphism. Another name for finite universal homeomorphisms is “almost isomorphisms,”so we will be able to say that the two schemes are almost isomorphic. This reduces thequestion of the difference between the GIT quotient and PsRd

R to a local question; will willtake up this question locally in Chapter 2.

From the previous paragraph, we know that the geometric points of the GIT quotient bythe adjoint action Rep,d

R //PGLd are in natural bijective correspondence with isomorphismclasses of semisimple representations of R. This naturality of this bijection refers to thecanonical map φ : RepdR → Rep,d

R //PGLd (resp. φ : RepdR → Rep,dR //PGLd) of (1.5.2.1), as

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each geometric fiber of φ (resp. φ) has a unique closed geometric point (cf. Remark 1.5.1.6)corresponding to a semisimple representation.

The map ψ : RepdR → PsRdR (resp. ψ : Rep

d

R → PsRdR) has a similar property: by

Theorem 1.3.1.1, the geometric points of PsRdR are in natural bijective correspondence with

isomorphism classes of semisimple representations, meaning that there is a unique semisimplepoint in each geometric fiber of ψ (resp. ψ). As ψ (resp. ψ) factors uniquely through ν :RepdR//PGLd in (1.5.2.2), ν is an isomorphism on geometric points.

From this we know that ν is finite type, radicial, and surjective. We recall this and a fewother useful basic definitions and properties.

Definition 1.5.4.1 ([Gro60, Definition 3.5.4]). Let f : X → Y be a morphism ofschemes.

(1) We call f radicial or, equivalently, universally injective, if for all fields k, the inducedmap of sets X(k)→ Y (k) is injective. As remarked in [Gro60, §3.5.5], it suffices toverify this property on algebraically closed fields.

(2) We call f a universal homeomorphism if after any base change by Y ′ → Y , fY ′ is ahomeomorphism. By [Gro67, Corollary 18.12.11], f is a universal homeomorphismif and only if it is integral, radicial, and surjective.

(3) We call f an almost isomorphism if f is a finite universal homeomorphism, or,equivalently, if f is a finite type universal homeomorphism.

To verify that ν is a universal homeomorphism, it remains to show that it is integral, or,equivalently, finite. This is what we show in the following

Theorem 1.5.4.2. Let S be an affine Noetherian scheme, and let R be a quasi-coherentfinitely generated OS-algebra. The map ν : Rep,d

R //PGLd → PsRdR induced by ψ or ψ is a

finite universal homeomorphism.

Remark 1.5.4.3. We must remark that there are well known facts that can be im-mediately applied to improve this theorem. Indeed, Chenevier has generalized in [Che11,Theorem 2.22(i)] (which we record below in Theorem 2.1.3.3) a result of Nyssen [Nys96] andRouquier [Rou96], showing that deforming an absolutely irreducible pseudorepresentation(recall Defintiion/Lemma 1.3.4.1) is equivalent to deforming the associated absolutely irre-ducible representation. This shows that ν is an isomorphism over this locus; this is alreadyvisible in the study of the locus that we already did in §1.3.4. However, we are deferringthis local study of pseudorepresentations to Chapter 2. These results on absolutely irre-ducible pseudorepresentations will be discussed in §2.1.3. The full extent of what we prove,which extends the result of Chenevier to the multiplicity free case, may be found in Theorem2.3.3.7.

We thank Brian Conrad for comments leading to this remark.

Remark 1.5.4.4. Let us remark on the basic difficulty in making Theorem 1.5.4.2 moreprecise than it currently is. That is, why is ν hard to control? One major issue is thatGIT quotients are not stable under base change. For example, if one could prove that ν hadgeometrically reduced fibers, then by [Gro67, Corollaire 18.12.6], ν is a closed immersion.

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Therefore we might take a geometric fiber of ν and ψ at a point D ∈ PsRdR(k),

Rep,dR ×PsRdR

Spec k

// Rep,dR

Repd

R ×PsRdRSpec k

// Repd

R

Rep,dR //PGLd ×PsRdR

Spec k //

Rep,dR //PGLd

Spec k // PsRdR.

Then we would want to draw conclusions about this fiber of ν by studying the fiber of ψ orψ. We could study the invariants of the action of PGLd on the upper left entry. However,non-flat base change of an adequate moduli space is no longer an adequate moduli space,but may differ from an adequate moduli space by an adequate homeomorphism [Alp10,Proposition 5.2.9(3)]. Adequate homeomorphisms are not necessarily reduced, so we areunable to conclude anything about the fiber of ν over D by considering the fiber of ψ orψ over D. In other words, the GIT quotient is not stable under base change, and the basechange of a GIT quotient may differ from the GIT quotient of a base change by an adequatehomeomorphism.

Remark 1.5.4.5. We expect that it follows from Procesi’s solution of the “embeddingproblem” for Cayley-Hamilton algebras9 in characteristic zero [Pro87] that we can show thatν is not only an finite universal homeomorphism, but an adequate homeomorphism. Theadditional content required is that ν is an isomorphism in characteristic zero.

Proof. As noted above, Kraft’s result (Theorem 1.5.3.1) implies that ν is surjective andradicial. It is also finite type, since the source and target of ν are each finite type over theNoetherian scheme S by Theorems 1.5.1.4(4) and 1.1.10.15, respectively. Then by [Gro67,Corollary 18.12.11], if in addition ν is finite, then ν is a finite universal homeomorphism.We will actually check that ν is universally closed in order to show that it is finite; this willsuffice because ν is clearly affine.

To show that ν is proper, we verify the valuative criterion for universal closedness. Usingour knowledge that this morphism is separated and finite type between Noetherian schemes,[LMB00, Theorem 7.10] (see also [Gro61a, Remark 7.3.9(i)]) allows us to verify the followingvaluative criterion on spectra of complete discrete valuation rings B with algebraically closedresidue fields: for every diagram

SpecK //

Rep,dR //PGLd

ν

SpecB // PsRdR

9The embedding problem for Cayley-Hamilton algebras (R,D) is the problem of finding an embedding ρ ofR into a matrix algebra M which is compatible with the native pseudorepresentation D, i.e. ρ induces amorphism in CHd(R) over D, i.e. D = det ρ.

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where K is the fraction field of B, there exists a field extension SpecK ′ → SpecK which isthe fraction field of a valuation ring B′ with a dominant map SpecB′ → SpecB such thatthere exists a section

(1.5.4.6) SpecK ′ //

SpecK //

Rep,dR //PGLd

ν

SpecB′ //

44

SpecB // PsRdR

In fact, we will achieve this where K ′/K is a finite field extension and B′ is the integralclosure of B in K ′.

Let D denote the pseudorepresentation of R over K associated to the K-point of PsRdR in-

duced by the B-point induced above, and let DB denote the underlying B-valued pseudorep-resentation, so that D = DB ⊗B K. Corollary 1.3.2.4(1) and its proof gives us a semisimple

representation ρ : R ⊗OS K ′ → Md(K′) in Rep,d

R (K ′) where K ′/K is a finite extension offields, and whose induced pseudorepresentation DK′ := det ρ appears as the base changefrom K to K ′ of the pseudorepresentation D. Sending the K ′-point ρ ∈ Repd,R (K ′) via φ

to an K ′-point of Repd,R //PGLd, this K ′-point lies over the K-point D of Repd,R //PGLdgiven in the data of the valuative criterion above. Taking B′ to be the integral closure of Bin K ′, we now have all of the maps of (1.5.4.6) except the desired diagonal section.

We claim that ρ is conjugate to the tensor by ⊗B′K ′ of a representation ρB′ : R⊗OSB′ →EndB′(L

′), where L′ is a rank d projective B′-module. The projection of this B′-point of

RepdR to Rep,dR //PGLd via φ is the desired section in (1.5.4.6). Therefore, proving the claim

will complete the proof of the theorem.To prove the claim, first note that the OS-algebra homomorphism R→Md(K

′) inducedby ρ factors through the Cayley-Hamilton algebra R → (R ⊗OS K)/CH(D) by Proposition1.2.4.3. Moreover, since D is induced by ⊗BK from DB ∈ PsRd

R(B), it factors throughR→ R⊗OS B/CH(DB), i.e. this map lies in the composite

R −→ (R⊗OS B)/CH(DB) −→Md(K′).

By Corollary 1.2.2.9, the B-algebra (R⊗OS B)/CH(DB) is finite as a B-module.Choose a d-dimensional K ′-vector space V ′ and choose a basis in order to draw an

isomorphism Md(K′)∼→ EndK′(V

′). Also choose a rank d B′-lattice L ⊂ V ′, where B′ isthe integral closure of B in the finite extension K ′/K. Now let L′ be the B′-linear span ofthe translates of L by R ⊗OS B. Since this is a finite B-module, L′ is a finite projectiveB′-submodule of V , which is therefore rank d. Its action of R⊗OS B′ induces ρ by applying⊗B′K ′. Now R ⊗OS B′ → EndB′(L

′) is an object of RepdR(B′) inducing ρ ∈ RepdR(K ′),completing the proof of the claim.

Here is a nice result of our work: the maps ψ and ψ are adequate moduli spaces up to analmost isomorphism. Some of the properties of an adequate moduli space still hold despitethis defect.

Corollary 1.5.4.7. With assumptions as in Theorem 1.5.4.2, the morphisms ψ and ψhave the properties (1), (2), and (4) proved of adequate moduli spaces in Theorem 1.5.1.4,as well as property (1) defining adequate moduli spaces in Definition 1.5.1.1. Namely, ψ andψ are finite type, universally closed, push forward coherent sheaves to coherent sheaves, and

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two geometric points in RepdR(k) (resp. Repd

R(k)) have overlapping closures if and only iftheir images under ψ (resp. ψ) are isomorphic.

Proof. We have shown that ν : Rep,dR //PGLd → PsRd

R is a finite universal homeomor-phism. Now we apply Theorem 1.5.1.4: because the canonical map φ from RepdR to the GITquotient is finite type and has the geometric point closure property, the same is true of thecomposition ν φ = ψ; because it is universally closed, its composition with the finite andtherefore proper map ν is still universally closed. Finally, since push forwards of coherentsheaves are coherent under φ and under the finite morphism ν, the same is true of ψ. This

all holds for ψ on Repd

R as well.

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CHAPTER 2

Local Study of Pseudorepresentations

In Chapter 1, we described moduli spaces of representations and pseudorepresentationsand proved that the maps ψ, ψ sending algebraic stacks of representations to their associatedpseudorepresentations are very close to adequate moduli spaces. In particular, they areuniversally closed. We accomplished this almost entirely through a study of the geometricpoints of these moduli spaces, the only additional input being the verification that ν satisfiesthe valuative criterion for properness in Theorem 1.5.4.2. However, as we noted in Remark1.5.4.4, the study of the defect ν of ψ (resp. ψ) from being an adequate moduli space isnot visible through the fibers of ψ (resp. ψ). The challenge is that the GIT quotient, whichis the base of adequate moduli space, does not admit a good moduli interpretation – onlyits geometric points have a satisfying moduli interpretation. However, as remarked at thebeginning of §1.5.4, we have reduced the study of the defect ν to ψ being an adequate modulispace to a local question on the base. This is one reason why we will now study the modulispace of pseudorepresentaitons locally. For example, in §2.3, we make progress in showingthat ν is an isomorphism by adding more linear structure to representations whose inducedpseudorepresentation deforms a fixed multiplicity free psueodrepresentation. In this case, wewill be able to eliminate the defect ν. This result is recorded in Theorem 2.3.3.7.

Of course, there are other reasons to study pseudorepresentations locally. One reasonis to study their tangent spaces and deformation theory, which is what we begin with,following Chenevier [Che11]. Our main result here is Proposition 2.1.2.3. This gives arepresentation theoretic condition for the finitude of the dimension of the tangent space to afield-valued pseudorepresentation. In this, we make an improvement on [Che11, Proposition2.28] by eliminating the assumption that the characteristic of the field must be larger thanthe dimension or must be 0. This follows from the application of PI ring theory describedin Chapter 1.

The other major goal in this chapter is to identify some projective subschemes of Repd

R,locally on the base PsRd

R of ψ. To accomplish this fiber-wise is to apply one of the resultsof King [Kin94] (Theorem 2.2.1.12 here), which shows that these projective spaces existinside geometric fibers of ψ. Our additional contribution is the deformation of this ampleline bundle to henselian neighborhoods of a point, so that the projective subscheme can bedeformed to complete local neighborhoods (Theorem 2.2.4.1). To this end, our work here isto carefully identify the ample line bundle implicit in King’s result.

Our motivating case of interest for this local study is the moduli of continuous pseudorep-resentations and representations of a profinite group or algebra, with a certain finitenesscondition. In this case, the results above apply very well, as the moduli formal scheme ofcontinuous pseudorepresentations is semi-local (see Corollary 3.1.6.13). Each component isthe formal spectrum of a complete local Noetherian ring! We are preparing the results inChapter 2 with their application to profinite representation theory in Chapter 3 in mind.

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2.1. Pseudorepresentations over Local Rings

In this section, we will study pseudorepresentations of an algebra R over a commutativelocal ring A. In practice, we will often fix a d-dimensional pseudorepresentation

D : R −→ A

and draw conclusions about D given some conditions about the data. We will begin withdeformation theory of a field-valued pseudorepresentation and then discuss the tangent spaceof the pseudorepresentation functor at such a point. We will conclude with some facts aboutCayley-Hamilton algebras (R,A) over local rings.

2.1.1. Deformation Theory Setup. Our study of the deformations of pseudorepre-sentations will follow Chenevier [Che11]. As usual, let A be a commutative ring and let Rbe an A-algebra. We could consider a closed SpecA-subscheme X ⊂ PsRd

R with its reducedstructure, and then study the completion of PsRd

R at X. However, our purposes do notrequire this generality; in particular, our work in Chapter 1 shows that we can study themorphism ν locally on the base. Our setting for the study deformations will be a completeNoetherian local base ring A with residue field FA of characteristic p ≥ 0, along with a givend-dimensional FA-valued pseudorepresentation of R, denoted

D : R⊗A FA −→ FA.For example, in this setting, A may be the Cohen ring of FA, which we denote by W .

We study deformations of D to the following rings, writing F for FA.

Definition 2.1.1.1. Let AF be the category of Artinian local A-algebras with residuefield F, where morphisms are local A-algebra homomorphisms.

Let AF be the category of Noetherian local A-algebras with residue field F, where mor-phisms are local W -algebra homomorphisms. For B ∈ AF we write mB for its maximalideal.

The category AF includes AF as a full subcategory, and objects in AF consist of limits(filtered projective limits with surjective maps) in AF.

We define the deformation functor PsRD as follows.

Definition 2.1.1.2. With the data p,A,R, D, d and F as above, let PsRD be the covari-ant functor on AF associating to each B ∈ ob AF the set of d-dimensional pseudorepresenta-tions

D : R⊗A B −→ B

such that D ⊗B F −→ F ∼= D. We call such deformations of D pseudodeformations.

The representability of this deformation functor in the category AF follows immedi-ately from the representability for the usual pseudorepresentation moduli scheme PsRd

R overSpecA.

Proposition 2.1.1.3. Given F, A,R, d, and D as above, let D also denote its associatedF-point of PsRd

R. Then the pseudodeformation functor is representable by the completion BD

of the local ring OPsRdR,Dat its maximal ideal mD, with the universal object Du ⊗ΓdA(R)ab BD.

Proof. By definition of pseudodeformation, any object of PsRD over B ∈ AD corre-sponds to a map SpecB → PsRd

R factoring through the natural map SpecBD → PsRdR.

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Corollary 2.1.1.4. If R is finitely generated as a A-algebra, BD is Noetherian.

Proof. In this case, PsRdR is Noetherian and finitely generated over SpecA by Theorem

1.1.10.15. Then since BD is the completion of a localization of a Noetherian ring, PsRD isNoetherian.

Since BD is a complete local ring, there are several conditions on BD equivalent to theNoetherian condition.

Lemma 2.1.1.5. Since BD is a complete local A-algebra and A is a complete Noetherianlocal ring, the Cohen structure theorem (see e.g. [MR10, Theorem 3.2.4]) implies that thefollowing properties are equivalent.

(1) There exists a surjection W [[t1, . . . , tn]] BD for some n ≥ 0.(2) There exists a surjection A[[t1, . . . , tn]] BD for some n ≥ 0.(3) BD topologically finite type1 as a A-algebra.(4) BD is Noetherian.(5) dimF(mD/m

2D

) is finite.(6) The tangent F-vector space PsRD(F[ε]/(ε2)) is finite-dimensional.

This is our motivation to study the tangent space TD := PsRD(F[ε]/(ε2)) of PsRdR at D.

We will give a (co)homological condition for the finiteness of the tangent space in the nextparagraph.

2.1.2. Tangent Spaces of the Pseudorepresentation Functor. We now describeand give a sufficient condition for the finiteness of the tangent space of the pseudorepresen-tation functor (or pseudodeformation functor) at a point. We follow Chenevier [Che11, 2.24-2.29] here. We make an improvement on Chenevier’s results, generalizing [Che11, Proposition2.28] to arbitrary characteristic in Proposition 2.1.2.3; the improvement entirely rests on theuse of the reference [Sam09] (see Theorem 1.2.2.6 and its use in Lemma 1.2.3.1), and wefollow Chenevier’s techniques otherwise.

This study is especially useful in preparation for giving sufficient conditions for theNoetherianess of a complete local “pseudodeformation ring” of continuous deformations ofa field-valued pseudorepresentation of a profinite algebra (see Theorem 3.1.5.3).

As usual, we have a commutative ring A and an A-algebra R. Let D : R → A be ad-dimensional pseudorepresentation. We will write A[ε] for A[ε]/(ε2), i.e. ε2 = 0. For anyA-module M , we write M [ε] for M ⊗A A[ε].

Definition 2.1.2.1. Let D be a d-dimensional pseudorepresentation D : R → A. Wecall a pseudorepresentation D : R[ε] → A[ε] a lift of D when D ⊗A[ε] A ∼= D. Through thecanonical identification

MdA(R,A[ε])

∼−→ PsRdR[ε](A[ε]),

of Corollary 1.1.3.10, the set of lifts is canonically functorially isomorphic to the set ofmultiplicative A-polynomial laws

P : R −→ A[ε]

such that they map to D via composition with the A-algebra homomorphism π = πA :

A[ε]ε7→0−→ A. We denote this set of multiplicative polynomial laws by T = TD ⊂Md

A(R,A[ε]),the tangent space at D.

1When we say that B is topologically finite type over A, we mean that it is (or admits a surjection from)the completion of some finite type A-algebra with respect to the powers of some ideal of A.

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Another way of defining this tangent space is to say that TD := (π∗)−1(D0), where

π∗ : HomA−alg(ΓdA(R)ab, A[ε]) −→ HomA−alg(Γ

dA(R)ab, A)

f 7→ π f.One can check that it has a natural A-module structure.

From now on, in our discussion of lifts of pseudorepresentations, we let A be a field F.

Lemma 2.1.2.2 ([Che11, Lemma 2.26]). Let R be a F-algebra and let D : R → F be ad-dimensional pseudorepresentation. Assume that there exists a positive integer N such thatker(D)N ⊂ CH(D). Then T ⊂ PdF(R/ ker(D)2N ,F).

Proof. Let P ∈ TD and let D : R[ε] → F[ε] be the associated pseudorepresentation.One can check that ker(P ) = ker(D)∩R. Therefore want to show that 2N such that satisfiesthe relation ker(D)2N ⊆ ker(D).

Consider the Cayley-Hamilton F[ε]-algebras S := R/CH(D) and S[ε], which is canoni-cally isomorphic to R[ε]/CH(D) by Lemma 1.1.8.6. For r ∈ ker(D)[ε] ⊂ R[ε], we have byassumption Λi(r) ∈ εF for all 1 ≤ i ≤ d, and therefore sd ∈ ε · F[s] for all s ∈ J [ε] ⊂ S[ε].Let J := ker(D)/CH(D) ⊂ S, so J ∼= J [ε]/εJ [ε]. The assumption ker(D)N ⊂ CH(D) impliesthat (J [ε]/εJ [ε])N = 0, and then J [ε]2N = 0. Consequently, ker(D)2N ⊂ CH(D) ⊂ ker(D),where the latter inclusion is the content of Lemma 1.2.1.1.

We continue to work with deformations of a fixed d-dimensional pseudorepresentationD : R→ F. Now let us restrict to the case that S := R/ ker(D) is finite-dimensional as a F-vector space. By Theorem 1.3.1.3, this condition will hold when F is perfect, p = char(F) > 0and [F : Fp] <∞, R/F is finitely generated, or d < p. This lemma improves [Che11, Lemmas2.26].

Proposition 2.1.2.3 (Following [Che11, Proposition 2.28]). Let R,F, D : R → F, andassume that S = R/ ker(D) is finite-dimensional over F. Then if Ext1

R(S, S) is finite-dimensional over F, where S is treated as a R-module here, then TD is also finite-dimensionalover F.

Compare this statement with Theorem 1.3.3.2; the methods of proof also correspond inlarge part.

Proof. Choose N such that ker(D) ⊂ CH(D). Such an N exists by Lemma 1.2.3.1(4),and is bounded by the integers N(d) and N(d, p) of Definition 1.2.2.7.

Now Lemma 1.3.3.1 tells us that

HomR(I/I2, S) ∼= HomS(I/I2, S)

is also finite-dimensional as a F-vector space. Since S is semisimple and the S-module S con-tains all simple S-modules as submodules, the finiteness of the dimension of HomS(I/I2, S)implies that I/I2 is finite length as a S-module, and therefore also implies that dimF I/I

2 <∞.

Because there are natural surjections

(I/I2)⊗nF In/In+1

for any ideal I ⊂ R, this means that R/ ker(D)2N is also finite-dimensional over F. UsingLemma 2.1.2.2, we know that TD ⊂ PdF(R/ ker(D)2N ,F). Finally, because the finitude of ageneral A-module M implies the finitude of ΓdA(M) as an A-module for any d ≥ 0, we applyTheorem 1.1.3.4 to conclude.

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We can apply Proposition 2.1.2.3 to give a criterion depending only on D for the Noethe-rianness of the complete local deformation ring BD defined in Proposition 2.1.1.3, using theNoetherianness criteria of Lemma 2.1.1.5. For this statement, we resume the language of§2.1.1, also setting R := R⊗A FA.

Corollary 2.1.2.4. Let A be a complete Noetherian local ring and let R be an A-algebra.Fix a d-dimensional pseudorepresentation D : R → FA such that if we take S := R/ ker(D)as an R-module, Ext1

R(S, S) is finite-dimensional as a FA-vector space. Then the completelocal pseudodeformation ring BD of Proposition 2.1.1.3 is Noetherian.

Since we will be interested in this primarily in the profinite topological case, we will givethe proof for the profinite case in Theorem 3.1.5.3. A proof in this case would feature thesame techniques without the topological considerations.

2.1.3. Cayley-Hamilton Pseudorepresentations over Local Rings. As we haveremarked, a d-dimensional Cayley-Hamilton A-algebra (R,D) shares properties with alge-bras appearing as subalgebras of d×d matrix algebras. For example, each element is integralof degree d over A, and if A is a field, we have shown that the Jacobson radical is nilpotent(see Lemma 1.2.3.1 and Corollary 1.2.2.10 for this and other properties of Cayley-Hamiltonalgebras). When A is a henselian local ring and the semisimple representation correspond-ing to the special fiber of D is absolutely irreducible and split over the residue field, thiscorrespondence with matrix algebras is exact. This is what we describe in this paragraph.

First we require a lemma on the Jacobson radical of Cayley-Hamilton algebras over localrings. Recall that J(R) denotes the Jacobson radical of the ring R.

Lemma 2.1.3.1 (Following [Che11, Lemma 2.10]). Let A be a local ring with maximal idealmA and residue field F = FA. Let R be an A-algebra with a d-dimensional Cayley-Hamiltonpseudorepresentation D : R→ A with residual pseudorepresentation D = D ⊗A F.

(1) The kernel of the canonical surjection R (R⊗A F)/ ker D is J(R).(2) If ms

A = 0 for s ≥ 1 an integer, then J(R)N(d)s = 0, where N(d) is the integer of Def-inition 1.2.2.7, which depends only on d. The possibly lesser integer N(d, charFA)can be used in place of N(d).

Our use of polynomial identity ring theory improves Chenevier’s result in the case d ≥charFA.

Proof. Write I for the two-sided ideal named in statement (1). Let us first show thatI ⊆ J(R), which will follow from checking that 1 + I ⊂ R×. By Lemma 1.2.3.1(1), itis equivalent to check that D(1 + I) ⊂ A×. But it is clear that D(1 + I) ⊆ 1 + mA byassumption, so we have I ⊆ J(R). To show the reverse inclusion, we first observe thatmA · R ⊆ I ⊆ J(R), so it will suffice to prove the reverse inclusion with A = FA. Now thedesired inclusion J(R) ⊆ I is given by Lemma 1.2.3.1(5).

Now we assume that msA = 0. It is clear that we may replace R by R/mA · R, assume

that A = FA, and show that J(R)N(d) = 0. This is precisely what we get from Lemma1.2.3.1(4).

Recall this essential property of henselian rings. The idempotent lifting is what we requirein order to make a comparison with a matrix algebra over all of SpecA, and not just overthe closed point.

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Lemma 2.1.3.2 (cf. [BLR90, §2.3, Proposition 4]). Let A be a local ring with residue fieldFA. Then A is Henselian if and only if for any finite A-algebra B, the canonical map onidempotents

Idem(B) −→ Idem(B ⊗A FA)

is an isomorphism.

Now we can give the main theorem of this paragraph.

Theorem 2.1.3.3 ([Che11, Theorem 2.22(i)]). Assume that D is Cayley-Hamilton andthat A is a henselian local ring with residue field FA. If D is split and absolutely irreducible,then there is an A-algebra isomorphism

ρ : R∼−→Md(A)

such that D = det ρ.

Proof. Omitted.

Recall the representation theoretic moduli spaces of §1.4. The local result Theorem2.1.3.3 is enough for us to show that the universal Cayley-Hamilton algebra is globally anAzumaya algebra when restricted to the absolutely irreducible locus PsIrrdR ⊂ PsRd

R. Wealso point out that ψ is an isomorphism over this locus, which immediately implies thatthe deformation functor of a chosen absolutely irreducible field-valued representation of Ris equivalent to the deformation functor of the representation. This is an improvement ofChenevier of the results of Nyssen [Nys96] and Rouquier [Rou96], who showed that deform-ing an absolutely irreducible pseudocharacter is equivalent to deforming the associated anabsolutely irreducible representation.

Corollary 2.1.3.4 ([Che11, Corollary 2.23]). Let A be a commutative ring and let Rbe an A-algebra.

(1) Over the absolutely irreducible locus PsIrrdR ⊂ PsRdR, the restriction of the universal

Cayley-Hamilton algebra E(R, d) to PsIrrdR is an Azumaya OPsIrrdR-algebra of rank

d2.(2) Over PsIrrdR ⊂ PsRd

R, ψ and ν are isomorphisms.(3) For each split point D ∈ PsIrrdR, the mD-adic completion of OPsIrrdR,D

is canonicallyisomorphic to the deformation ring for the representation

R −→Md(κ(D)).

Proof. Chose x ∈ PsIrrdR and let B be the strict henselization of OPsRdR,x. By Lemma

1.1.8.6,E(R, d)⊗O

PsRdR

B∼−→ (R⊗A B)/CH(Du ⊗ A).

Theorem 2.1.3.3 now implies that the right hand side is isomorphic to Md2(B). HenceE(R, d) ⊗O

PsRdR

OPsRdR,xis an Azumaya algebra of rank d2 since OPsRdR,x

→ B is faithfully

flat (cf. [Sta, Lemma 07QM]). Now we observe that E(R, d) is an Azumaya algebra, as thedefinition of an Azumaya algebra may be given locally (see Definition 1.4.1.5(1)).

Parts (2) and (3) follow at once, as the Azumaya OPsIrrdR-algebra defines a section to ψ

over PsIrrdR.

Remark 2.1.3.5. This generalizes results previously known for pseudocharacters whenthe characteristic is larger than the dimension, e.g. [Nys96, Rou96, Car94]. In particular,

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Carayol [Car94] showed that a deformation of an absolutely irreducible residual representa-tion is characterized by its trace. Nyssen and Rouquier [Nys96, Rou96] showed the “con-verse,” that the deformation of a residual pseudocharacter arising as the trace of an absolutelyirreducible representation is realizable as the trace of a deformation of said representation.

We have succeeded in showing that ψ is an adequate moduli space over the absolutelyirreducible locus, but this is a trivial case since ψ is an isomorphism here. We will prove thisin a nontrivial case in Corollary 2.3.3.9.

2.2. Fibers of ψ

Recall Theorem 1.4.3.1, where we show that schemes and stacks parameterizing d-dimensional representations of an algebra R are equivalent to the analogous moduli space forrepresentations of the universal Cayley-Hamilton algebra E(R, d) over the universal pseu-dorepresentation of R. This is a particularly useful result in the case that E(R, d) is finiteas a OPsRdR

-module. We have shown that this is true when, for example, A is Noetherian

and R is finitely generated (Corollary 1.4.3.3).Assuming that E(R, d) is finite, we study of the fibers of ψ (resp. ψ, resp. ψ). Fix a

residue field F of PsRdR and let D denote the associated pseudorepresentation

D = Du ⊗ F : E(R, d)⊗ΓdA(R)ab F −→ F.Recall that by Lemma 1.1.8.6, the formation of the Cayley-Hamilton quotient of R commuteswith base change over PsRd

R. Therefore, when E(R, d) is finite over OPsRdR, the study of the

fibers of ψ amounts to the study of representations of a finite-dimensional algebra

E(R, d)⊗ΓdA(R)ab Fover the field F, with the condition that the induced pseudorepresentation of these represen-tations is precisely D.

Consider also the case where R is an algebra over a field F that is not finitely generated,but where a pseudorepresentation D : R→ F satisfies the conditions of Theorem 1.3.3.2, sothat the associated Cayley-Hamilton algebra E := R/CH(D) is finite-dimensional over F.Then, using the universality of the Cayley-Hamilton algebra (Theorem 1.4.3.1), the repre-sentations of R inducing D as a determinant amount to the representations of E inducingD|E as a determinant. Once again, we are reduced to the study of the representations of afinite-dimensional algebra. We will also find ourselves remanded to this case when we studyrepresentations of profinite topological algebras in Chapter 3, provided that an appropriatefiniteness condition is satisfied.

Therefore, for this section we will let E be a finite-dimensional F-algebra with a givenCayley-Hamilton d-dimensional pseudorepresentation

D : E → F.Certainly, this satisfies the conditions of Theorem 1.4.1.3, so that the scheme of framedrepresentations and the algebraic stacks of representations are finite type SpecF-schemes.We will study the fiber of the representation spaces of E over D, i.e.

Rep,D

:= ψ,−1(D) ⊂ Rep,dE ,

RepD := ψ−1(D) ⊂ RepdE,

RepD := ψ−1(D) ⊂ Rep

d

E.

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Important point. This condition that a representation lie in the fiber of D is equivalent,once D is split, to the condition that its Jordan-Holder factors match those of the semisimplerepresentation ρss

Dassociated to ρ via Theorem 1.3.1.1. Therefore, RepD is a geometric real-

ization of the category (whose morphisms are isomorphisms) of representations of E with agiven semisimplification. Of course, semisimple representations of a finite-dimensional alge-bra are naturally in bijective correspondence with functions from simple representations tothe non-negative integers. This description is known as a dimension vector. Therefore, onceF is large enough so that E/ ker(D) is split, we can speak of F-valued pseudorepresentationsas dimension vectors, and vice-versa. This lange will be particularly natural as we introducerepresentations of quivers.

The main goal of our study is to show that there are projective subspaces of RepD

corresponding to certain notions of (semi)stability, formally analogous to the theory of vectorbundles over a curve. Basically, we are reviewing a result of A. D. King [Kin94]. Let usbegin with a brief summary of his result. We will freely use terminology from §1.3.4.

Given an integer-valued character θ : K0(RepE(F)) → Z of the Grothendieck group ofE, he develops a corresponding notion of semi-stability and stability for representations ofE. He then shows that semistability (resp. stability) of a representation ρ ∈ Rep

D ⊂ Rep,dE

is equivalent to it lying in a semistable (resp. stable) orbit for a certain action of a certainreductive group and a linearization of Rep

E corresponding to θ. Then the GIT quotient of thesemistable orbit locus is a projective space which is a coarse moduli space parameterizing θ-semistable representations of E up to S-equivalence. The notion of S-equivalence is analogousto the notion of S-equivalence of vector bundles on curves due to Seshadri [Ses67]. Theequivalence relation is better on the stable locus within the GIT quotient: it is a coarsemoduli space for θ-stable representations with respect to the usual notion of equivalencebetween E-modules.

We will not pursue these generalities and the notion of S-equivalence. Rather, we willfocus on a particular case when we get a projective, fine moduli space out of this GITconstruction. This case is noted by King [Kin94, Remark 5.4]: θ may be chosen (relative toD) so that θ-semistability implies θ-stability in Rep

D, and such that the GIT quotient is a fine

moduli space.2 This will show that there are large projective subschemes of RepD(θ) ⊂ Rep

D

corresponding to θ-(semi)stable representations of E.

Remark 2.2.0.6. This observation generalizes and answers affirmatively a suspicion ofKisin [Kis09a, Remark 3.2.7] on the existence of projective loci (relative to ψ) inside modulispaces of representations, and adds many more instances to the cases that Kisin pointed out(see Corollary 2.2.2.14). Of course, we must wait until Chapter 3 to see that the case ofextensions of continuous representations of a profinite group with finite field coefficients canbe reduced to the case that we now work with.

In terms of the intrinsic study of the special fiber of ψ, King’s result is all that we require,and it would suffice to quote his result in order to show which families of representations ofE form projective spaces in Rep

D. However, we are also interested in showing that these

projective sub-moduli-spaces exist locally on the base in PsRdR, or complete-locally in the

profinite case PsRD ⊂ PsRdR. Of course, this will follow if we can show that an ample

line bundle for King’s projective space is the specialization of a locally well-defined line

bundle in Repd

R to PsRdR. We will accomplish this over complete local rings, with our work

2The former condition is the more interesting one.

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culminating in Theorem 2.2.4.1. This will take some work, since King’s work uses the factthat the category of modules for any finite-dimensional algebra is equivalent to the categoryof modules for a a quotient algebra of a path algebra for a finite quiver. It is in the categoryof representations of quivers that these projective spaces are most naturally constructed,and our work is to follow the ample line bundle on a space of representations of a quiverthrough several equivalences necessary to identify an ample line bundle on a certain spaceof representations of E.

Assumption. We assume that F is algebraically closed. This assumption will be inplace only for this section.

Remark 2.2.0.7. This assumption is used to ensure that R/ ker(D), the semisimplealgebra associated to the pseudorepresentation, will be split in the sense of Definition 1.3.4.4.It is also necessary in order to ensure that statements about points of a GIT quotient areaccurate, as GIT only has a good functor of geometric points (cf. Remark 1.5.1.6). Theformer issue is more serious, as we will need to find as many idempotents as the dimensionof an algebra in order to draw comparisons with quivers. In many cases, including thosethat we will be concerned with for profinite algebras in Chapter 3, this can be achieved witha finite separable extension of a field F. Therefore an assumption that D is split over F willbe sufficient to apply the results of this section.

2.2.1. King’s Result on Quiver Representation Moduli. We give a brisk intro-duction to quivers in order to state King’s result. For more background on quivers, see forexample [ASS06].

Definition 2.2.1.1. A quiver Q is an oriented graph Q = (Q0, Q1), where Q0 is the setof vertices, and Q1 the set of oriented edges, also known as arrows. We define the head andtail functions

h, t : Q1 → Q0

to be the maps sending an arrow a ∈ Q1, to the head h(a) of the arrow and the tail t(a) ofthe arrow. A quiver Q is called finite if Q0 and Q1 are finite.

Definition 2.2.1.2. Let Q be a quiver.

(1) A representation of Q over a field F is a collection of F-vector spaces Wv for eachv ∈ Q0 and a collection of F-linear maps φa : Wt(a) → Wh(a) for each arrow a ∈ Q1.

(2) A morphism of such representations, (Wv, φa)→ (Uv, ψa) is a collection of F-linearmaps fv : Wv → Uv such that fh(a) φa = ψa ft(a) for each a ∈ Q1.

(3) The dimension vector β ∈ ZQ0 of a representation (Wv, φa) is the vector of integersβv = dimFWv for each v ∈ Q0. A representation is called finite-dimensional if Wv

is finite-dimensional for all v and βv = 0 for all but finitely many v ∈ Q0.(4) Given a dimension vector β we use GL(β) to denote the group ×v∈Q0GL(Wv) of

linear automorphisms of (Wv).(5) ∆ ⊂ GL(β) denotes the diagonal subgroup of scalars (t, . . . , t) ⊂ GL(β), and

PGL(β) denotes the quotient.

Note that PGL(β) is not generally the product over v ∈ Q0 of PGL(Wv).Convention. We will work with finite quivers and finite-dimensional representations

from now on, without remarking on their finiteness.

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Once we define the path algebra FQ of Q, we will see that under the equivalence ofbetween representations of Q and representations of FQ, dimension vectors correspond topseudorepresentations of FQ.

First, we note that framed moduli spaces of representations of quivers are affine spaces!Indeed, the set of representations of F over a given dimension vector β ∈ ZQ0 correspondingto the set of vector spaces (Wv)v∈Q0 is

Repβ (F) :=

⊕a∈Q1

HomF(Wt(a),Wh(a)).

The group GL(β)(F) acts naturally on this set, and one can check that two representationsin Rep

β (F) are isomorphic if and only if they lie in the same orbit of GL(β)(F).

We let Repβ represent the functor from SpecF-schemes to the set of such representations;

explicitly, this functor sends a SpecF-scheme X to the OX-module⊕a∈Q1

HomOX (Wt(a) ⊗F OX ,Wh(a) ⊗F OX).

Observe that there is a natural isomorphism

(2.2.1.3) Repβ∼−→ Spec Sym∗F

(⊕a∈Q1

HomF(Wt(a),Wh(a))∧

),

and that the algebraic group GL(β) acts naturally on Repβ , with orbits consisting of iso-

morphism classes of representations. In addition, PGL(β) acts on Repβ ; it acts on each

space Hom(Wt(a),Wh(a)) even though it does not have a sensible action on Wv for v ∈ Q0.In analogy to Definition 1.4.1.1, we define the following groupoids of representations.

Definition 2.2.1.4. Let Q be a quiver. Define groupoids on SpecF-schemes by mappingan SpecF-scheme X to the following sets.

Repβ := X 7→ For each v ∈ Q0, a vector bundle Wv/X of rank βv,

for each a ∈ Q1, φa ∈ HomOX (Wt(a),Wh(a)).

The definition of Repβ amounts to tracking the data of the the data of (Wv, φa) modulo

simultaneous twists of (Wv) by a line bundle L ∈ PicX.3

Repβ := X 7→ For v, w, x ∈ Q0, a vector bundle Hvw/X of rank βvβw,

OX-Azumaya algebra structure on Ev := Hvv,

OX-module surjections cvwx : Hvw ⊗OX Hwx → Hvx,

and, for a ∈ Q1, φa ∈ Ht(a)h(a)such that the following conditions on cvwx hold (following [BC09, §1.3.2]):

(UNIT) For all v, w ∈ Q0, cvvw : Ev ⊗ Hvw → Hvw (resp. cvww : Hvw ⊗ Eww → Hvw) iscompatible with the Azumaya algebra structure on Ev.

(ASSO) For all v, w, x, y ∈ Q0, the two natural maps Hvw ⊗Hwx ⊗Hxy → Hvy coincide.(COM) For all v, w ∈ Q0, x ∈ Hvw, y ∈ Hwv, cvwv(x⊗ y) = cwvw(y ⊗ x).

3This corrects a small oversight in [Kin94] – he does not mention the twists of families of representations ofQ by a line bundle and the resulting lack of representability of Repβ .

89

In analogy to Theorem 1.4.1.4, one can check that there are natural equivalences ofSpecF-groupoids

(2.2.1.5) Repβ∼−→ [Rep

β /GL(β)], Repβ∼−→ [Rep

β /PGL(β)].

Now choose θ ∈ ZQ0 , which we call a character of Rep(Q). In fact, such a characternaturally determines a character of the Grothendieck group of Rep(Q) (cf. Definition 1.3.4.5).Simple Q-representations over F are in natural bijective correspondence with Q0, sendingw ∈ Q0 to the representation (Wv, φa), where we have

Wv =

Wv = F v = wWv = 0 v 6= w

and we have φa = idWv if h(a) = t(a) = v, and φa = 0 otherwise. This establishes anatural equivalence between characters of Rep(Q) characters of the Grothendieck groupK0(RepQ(F)). We call a character of Rep(Q) indivisible if it is not the scalar multiple ofanother character.

We will define two notions of θ-semistability (resp. θ-stability), one intrinsic to the rep-resentation theory, and one being that of the GIT notion of semistability (resp. stability)of a point in Rep

β for the χθ-linearized action of GL(β), where χθ is a character of GL(β)associated to θ.

First we give the representation theoretic definition for a general F-algebra E, whichmakes sense for representations of a quiver Q even though we have not yet realized therepresentations of Q as the representations of its path algebra. As King points out, thisdefinition makes sense for any abelian category; special cases of the notion include Mumford’snotion of stability for vector bundles over a curve.

Definition 2.2.1.6 ([Kin94, Definition 1.1]). With F, E, and a character

θ : K0(RepE)→ Zas above,

(1) a representation W ∈ RepE is called θ-semistable if θ(M) = 0, and for every sub-representation W ′ ⊆ W , θ(W ′) ≥ 0.

(2) if W ∈ RepE is θ-semistable, and if, additionally, it satisfies the property

θ(W ′) = 0 =⇒ W ′ = W or W ′ = 0

for all subrepresentations W ′ ⊆ W , then we call W θ-stable.

We call two θ-semistable representations S-equivalent if they have identical compositionfactors in the full abelian subcategory of θ-semistable representations; the stable represen-tations are the simple objects in this subcategory. We will not focus here on S-equivalenceexcept when it coincides with the usual notion of equivalence.

In fact, as we pointed out in the introduction, we are mainly interested in families ofrepresentations in which θ-semistability implies θ-stability. As connected families of rep-resentations of finite-dimensional algebras have constant residual pseudorepresentation (byTheorem 1.1.7.4(5), for example), this condition is dependent upon the semisimplification ofthe representation. Semisimple representations amount to non-negative integer-valued linearcombinations of simple representations and simple representations are a basis for K0(RepE).We recall that this is the dimension vector (Definition 1.3.4.5) of the representation. Notethat the condition θ(W ) = 0 of semisimplicity depends only on its dimension vector θ, and

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can be expressed in terms of the dot product of the dimension vector with θ, i.e.

〈β, θ〉 = 0.

Using this terminology, we will give a condition such that semistability will imply stability.First, we require the following definition.

Definition 2.2.1.7. A standard projection operator on characters of K0(RepE) sendsθ ∈ K0(RepE) to its projection along the submodule spanned by a subset of the simplerepresentations of E. We say that a standard projection operator is non-trivial on thesupport of β ∈ K0(RepE) provided that Pβ 6= β.

Lemma 2.2.1.8. Let θ be a character of the Grothendieck group K0(RepE) of RepE, andlet β be a dimension vector such that 〈β, θ〉 = 0. If for every standard projection P thatis non-trivial on the support of β we have a strict inequality 〈Pβ, Pθ〉 6= 0, then for everyrepresentation W ∈ RepE with dimension vector β, W is θ-semistable if and only if it isθ-stable.

Definition 2.2.1.9. If β and θ satisfy the conditions of Lemma 2.2.1.8, we say that β isstabilizing with respect to θ.

Example 2.2.1.10. Let ρ1, . . . , ρn be simple representations of E, possibly with multi-plicity except that we demand that ρn 6' ρi for 1 ≤ i < n. Let β be the dimension vectorsupported on the ρi, with these multiplicities. Later in Example 2.2.3.1, we will study thisdimension vector relative to the character θ on K0(E) sending

θ :ρi 7→ 1 1 ≤ i < nρn 7→ −(n− 1)

We see that β is stabilizing with respect to θ. The only way to get a sum of zero out ofa projection to some subset of the isomorphism classes of the ρi is to choose the identityprojection.

Now we give a character of GL(β) associated to θ.

Definition 2.2.1.11. For each v ∈ Q0, write detv for the determinant of the vth com-ponent of GL(β) ∼= ×v∈Q0GL(Wv). Then set χθ to be the character

GL(β) −→ Gm

(gv) 7→∏v∈Q0

detv

(gv).

This geometric notion of semistability (resp. stability) cuts out a subfunctor of Repβ ,

which geometric invariant theory implies is open. We write

Rep,sβ (θ) ⊂ Rep,ss

β (θ) ⊂ Repβ

for these open subschemes. LetRep,ss

β (θ)× L(η)

denote the total space of the trivial line bundle L over Rep,ssβ (θ) with an action of GL(β)

extended to this space by acting on L by η−1 with the character χ−1θ . Then standard GIT

results give us that a quotient by GL(β) exists. This linearized GIT quotient space is

Repβ //(GL(β), χθ) := ProjF

(⊕n≥0

F[Rep,ssβ (θ)× L(χnθ )]GL(β)

),

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and it is projective over the GIT quotient

Repβ //GL(β) ∼= SpecF.

We writeRep

ss

β (θ) := Repβ //(GL(β), χθ)

for this projective SpecF-scheme, and standard GIT theory gives an open F-subscheme

Reps

β(θ) ⊆ Repss

β (θ)

image of the χθ-stable locus in Repβ (θ).

This theorem summarizes the GIT content of King’s paper.

Theorem 2.2.1.12 ([Kin94, Propositions 3.1-3.2, Proposition 5.2-5.3]).

(1) A point in Repβ corresponding to a representation W of Q is χθ-semistable, i.e. lies

in Rep,ssβ (θ) (resp. χθ-stable, i.e. lies in Rep,s

β (θ)), if and only if W is θ-semistable(resp. θ-stable).

(2) Two θ-semistable representations correspond to points in Rep,ssβ (θ) with GL(β)-

orbits with overlapping Zariski closures in Rep,ssβ (θ) if and only if the representa-

tions are S-equivalent with respect to θ.(3) Rep

ss

β (θ) is a coarse moduli space for families of θ-semistable modules up to S-equivalence

(4) When the dimension vector β is indivisible, the stable quotient Reps

β(θ) is a finemoduli space for families of θ-stable modules.

Much the content of King’s paper works toward proving Theorem 2.2.1.12 in the contextof representations of a finite-dimensional F-algebra, using an equivalence between quiverrepresentations with a fixed dimension vector on one hand, and representations of a finite-dimensional F-algebra E with induced pseudorepresentation D on the other. However, weneed a more concrete realization of this equivalence than King provides. We are stayingwithin the context of quiver representations for the moment so that we can carefully iden-tify an ample line bundle on Rep

ss

β (θ). Our goal is to give an explicit translation betweenrepresentations of Q and representations of E, also translating conditions of semistability,etc., so that under the resulting closed immersion

Repss

E,D(θ) → Repss

Q,β(θ),

we can identify the pullback of the ample line bundle to the left side in terms of the moduliproblem there. Therefore, let us conclude our overview of King’s results by identifying thisample line bundle on the right side in terms of the moduli problem there.

An ample line bundle on the linearized GIT quotient

Repss

β (θ) = ProjF

(⊕n≥0

F[Rep,ssβ (θ)× L(χnθ )]GL(β)

)is the standard ample line bundle Oθ(1) of this Proj construction,4 which consists of the

regular functions on Rep,ssβ (θ) × L(χnθ ) such that GL(β) acts by χθ. By reviewing the

definition of χθ and the explicit form of the coordinate ring for Repβ in (2.2.1.3), we observe

4It is not necessarily very ample.

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that this line bundle is the descent of the PGL(β)-equivariantly linearized line bundle

(2.2.1.13) Oθ(1) :=⊗v∈Q0

det(Wv)⊗θ(v)

on Repβ to Rep

ss

β (θ). In saying that this bundle is PGL(β)-equivariantly linearized, we

are using the fact that the GL(β)-linearization of Oθ(1) descends to a PGL(β)-linearization(cf. the comments on the action of PGL(β) on Rep

β at (2.2.1.3)). This is the case becauseχθ(∆) = 1; indeed, this is a condition for (GIT) semistability, which, in the translationbetween the representation theoretic and GIT notions of semistability for a representationW , corresponds to the condition θ(W ) = 0. Alternatively, we know that Oθ(1) will descendto Repβ by (2.2.1.5), and one can check that in terms of the intrinsic definition of Repβ,

(2.2.1.14) Oθ(1) ∼=⊗

v,w∈Q0

(∧βvβwHvw)⊗n(v,w)

for appropriate integers n(v, w) dependent on θ ∈ ZQ0 and β. We derive these integers from(2.2.1.13) by recalling that the natural association is Hvw = Hom(Wv,Ww), and

∧βvβwHvw∼= (∧βvWv)

⊗−βw ⊗ (∧βwWw)⊗βv .

Indeed, the integers n(v, w) are specified by the following

Lemma 2.2.1.15. Let β = (β1, . . . , βn) ∈ Zn be indivisible, and let θ = (θ1, . . . , θn) ∈ Znsuch that the dot product β · θ is zero. Then θ is a Z-linear combination of the vectors eij

for 1 ≤ i < j ≤ n, where eij = (eij1 , . . . , eijn ) is given by

eijk =

+βj if k = i−βi if k = j0 otherwise

.

The integers n(v, w) are the coefficients of evw in the expression for θ. Of course, sinceevw = −ewv for any v, w ∈ Q0, we don’t lose anything by restricting to i < j. Also, notethat we are assuming that βv > 0 for each v ∈ Q0.

Proof. We write Zn for the Z-module of n-tuples of integers. The standard dot productdefines a perfect pairing Zn × Zn → Z, and pairing with β defines a Z-module morphismµ = 〈·, β〉 : Zn → Z. The indivisibility of β is equivalent to the surjectivity of µ. Thereforeif we let M represent the kernel of µ, we have an exact sequence

0 −→M −→ Zn −→ Z→ 0

which admits a splitting. We observe that M must be free of rank n− 1 over Z. We want toverify that the sub-Z-module of M generated by eij, 1 ≤ i < j ≤ n, which we will denote byN , is in fact equal to M . Let P denote the cokernel of N → M , so that we have an exactsequence

0 −→ N −→M −→ P −→ 0.

Since e1j, 2 ≤ j ≤ n, are linearly independent over Q, N is free of rank n− 1 and P is finitein cardinality. We will complete the proof by showing that P ⊗Z Fp = 0 for all rationalprimes p, and we will do this by showing that eij span M modulo p for each p.

Fix a prime p. Because β is indivisible, there exists some v ∈ Q0 such that βv 6≡ 0(mod p). Without loss of generality, assume v = 1. Then the n− 1 elements e1j are linearly

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independent modulo p. This shows that the image of N⊗ZFp in M⊗ZFp is n−1-dimensional,and therefore is equal to M ⊗Z Fp.

2.2.2. Finite Dimensional Algebras and Path Algebras. Now we prepare back-ground material on finite-dimensional algebras to show that their representations can beexpressed as representations of quivers.

One step will be to show that given any algebra, its abelian category of representationsis equivalent to that of some basic algebra.

Definition 2.2.2.1. Let E be a finite-dimensional F-algebra.

(1) We call E basic provided that it has a complete set e1, . . . , en of primitive or-thogonal idempotent such that Eei and Eej are not isomorphic as F-algebras for alli 6= j.

(2) We call E connected provided that it cannot be written as a proper product ofalgebras E ∼= E1 × E2.

One can show (cf. [ASS06, Proposition I.6.2]) that the simple representations of a basicalgebra are all one-dimensional, or, equivalently, that if E is basic, then

(2.2.2.2) E/J(E) ∼= Fn, some n ≥ 0.

If e1, . . . , en are a complete set of primitive orthogonal idempotents for E, then eachsimple representation into F is given by sending ei to 1 for a single i, and J(E) and theremaining ej, j 6= i, to 0. Therefore, pseudorepresentations of a basic algebra E are inbijective correspondence with n-tuples of non-negative integers, where we have numbered acomplete set of n primitive idempotents correspondingly.

Next, we describe the path algebra FQ of a quiverQ. This is a basic algebra whose abeliancategory of representations is naturally equivalent to the abelian category of representationsof a given quiver Q. We will describe this equivalence below.

Definition 2.2.2.3. Let Q0 be a finite quiver.

(1) The path algebra FQ of Q is the quotient of the free algebra on the set Q0 ∪ Q1

where we write εv for v ∈ Q0 and αa for a ∈ Q1, subject to the relations

εvεw = δvwεv, εvαa = δvt(a)αa, αaεv = δh(a)vαa,

αaαb = 0 if h(a) 6= t(b),∑v∈Q0

εv = 1.

(2) Let J(FQ) be the Jacobson radical of FQ, which one can check is generated by thearrows. We call I ⊂ FQ an admissible ideal provided that there exists m ≥ 2 suchthat J(Q)m ⊆ I ⊆ J(Q)2.

We observe that FQ is a basic algebra.

Remark 2.2.2.4. There is another sensible definition of FQ when Q is not finite, ex-pressing FQ as ring graded by the lengths of paths. However, this definition does not havea unit when Q0 is infinite, and is equivalent to the definition given above when Q is finite.

Now we give a construction of a quiver from a connected basic algebra. This is an inverseconstruction to the construction of the path algebra.

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Definition 2.2.2.5 (cf. [ASS06, Definition II.3.1]). Let E be a basic connected F-algebra.Number off the complete set of primitive orthogonal idempotents e1, . . . , en. Now definethe (ordinary) quiver QE = (Q0, Q1) of E by Q0 = v1, . . . , vn in correspondence with theidempotents, and each arrow a in Q1 consists of a head h(a) = vi, tail t(a) = vj, and anelement of a fixed F-basis of ei(J(E)/J(E)2)ej.

The term “ordinary” is not standard notation, but we mention it as there are otherquivers associable to E such as its Auslander-Reiten quiver. We observe that if E is finite-dimensional, then Q is finite. One can check that Q does not depend on the choices made inorder to construct it. Also, given a primitive idempotent e ∈ E, we will often simply writeve for the vertex of Q associated to e.

Theorem 2.2.2.6 (cf. [ASS06, Theorem II.3.7]). Let E be a basic connected F-algebra.Then there exists a surjection of F-algebras from the path algebra of a connected quiver,namely, from the path algebra of its ordinary quiver

FQE −→ E,

inducing an isomorphism of E with the quotient QE/I of FQE by an admissible ideal.

Proof. Map the set Q0 into FQE by sending v to εv. By definition, an arrow a ∈ Q1

such that t(a) = v and h(a) = w is an element of some basis for ew(J(E)/J(E)2)ev. Choosea lift of this basis element to J(E) and map the set Q1 to FQE according to the choicesabove. This map is, in fact, surjective with admissible kernel [ASS06, Theorem II.3.7], andwe can already see that the kernel is contained in J(E)2.

Let us explicitly describe an equivalence between representations of Q = QE and repre-sentations of the path algebra FQ. We will give the construction a representation of FQ outof a representation of Q in terms of the algebraic stacks

RepQ,β∼−→ RepFQ,Dβ ,

since we are interested in keeping track of the line bundle Oθ(1) of (2.2.1.14) on RepQ,βafter its pullback to RepFQ or RepE. Here Dβ denotes a pseudorepresentation of FQ thatcorresponds via Theorem 1.3.1.1 to the direct sum of representations⊕

v∈Q0

M⊕βvv ,

where Mv is the one-dimensional simple representation on which only ev acts as 1 andew, w 6= v and Q1 act as 0. This is the semisimple representation of FQ associated to the|Q0|-tuple (ev)v∈Q0 , i.e. the direct sum with multiplicity (ev)v∈Q0 over the |Q0| simple (1-dimensional) representations of FQ corresponding to ev, cf. (2.2.2.2). For this constructionin the more usual framed setting (i.e. for elements of Rep

Q,β), see e.g. [ASS06, TheoremIII.1.6].

Let (Hvw, φa) ∈ Repβ(X) be a representation of Q over X ∈ SchSpecF of dimension vectorβ, as in Definition 2.2.1.4. Define

(2.2.2.7) E = E(Wv, φa) :=⊕

v,w∈Q0

Hvw.

The structure maps cvwx of Definition 2.2.1.4 endow E with the structure of an AzumayaOX-algebra, where the OX-algebra structure map induced by the sum over v ∈ Q0 of the

95

maps OX → Ev ∼= Hvv. One can readily check that the map of sets

Q = Q0 ∪Q1 −→ Eεv 7→ idv ∈ Ev ⊂ Eαa 7→ φa ∈ Ht(a)h(a) ⊂ E .

extends to a homomorphism FQ⊗FOX → E of OX-algebras. Here Ev and Ht(a)h(a) are beingconsidered as (OX-module) summands of E . Let Dβ : FQ→ F be the pseudorepresentationassociated to the semisimplification FQ →

⊕v∈Q0Ev, which one can check from the defini-

tions. We have constructed a map of SpecF-groupoids RepQ,β → RepFQ,Dβ ; one can examine

the inverse construction (cf. [ASS06, Theorem III.1.6]) to see that this is an equivalence, anduse e.g. [Kin94, Proposition 5.2] to show that the map is algebraic.

Now let E be a basic connected F-algebra. Choose a (non-canonical) surjection FQE Edescribed in the proof of Theorem 2.2.2.6. Choose also a Cayley-Hamilton pseudorepresen-tation Dβ : E → F, where we let β denote the dimension vector of Q corresponding to thesemisimple representation of FQ induced by the semisimple representation ρss

Dβ. We have a

closed immersionRep

E,Dβ→ Rep

FQ,Dβ∼−→ Rep

Q,β,

where the maps are equivariant for the natural action of PGL(β). Therefore we have a closedimmersion of algebraic stacks

RepE,Dβ → RepFQ,Dβ∼−→ RepQ,β,

and we observe that the line bundle Oθ(1), expressed in terms of the data Hvw on RepQ,β in(2.2.1.14), pulls back via the association (2.2.2.7) to a line bundle that can be constructed

out of appropriate sub-modules of the universal Azumaya algebra E on RepE,Dβ

receiving

the universal representation of E with pseudorepresentation Dβ.Now, in the case of basic algebras, we have achieved our goal of identifying the line bundle

Oθ(1). Let us extend this to the general case of a finite-dimensional algebra E.First we explain how to associate a basic algebra to a general F-algebra.

Definition 2.2.2.8. Let E be a finite-dimensional F-algebra with a complete set ofprimitive orthogonal idempotents e1, . . . , en. Partition these idempoents according to the

equivalence relation ei ∼ ej if and only if there is a F-algebra isomorphism Eei∼→ Eej, and

choose a representatives ej1 , . . . , ejb . Write eE for eE =∑b

1 eji . Then the F-subalgebra

Eb := eEEeE

as a basic algebra associated to E.

This subalgebra Eb of E is clearly not canonical, but one can show that the isomorphismclass of Eb does not depend on the choices above. For the time being, we fix such choicesand the resulting F-subalgebra Eb ⊆ E.

Remark 2.2.2.9. For this association to work, we must have a complete set of orthogonalidempotents as the definition above requires. When we apply this construction, we willalways work with finite-dimensional algebras E whose semisimple quotient E/J(E) by theJacobson radical is a product of matrix algebras. Such a set of idempotents clearly exists

96

for a product of matrix algebras, and we then apply the well known fact that one can (non-canonically) lift idempotents over a nilpotent ideal (the Jacobson radical of a finite dimensionalgebra is nilpotent).

Here we see that we can reduce the study of representations of finite-dimensional algebrasto the study of the representations of basic algebras.

Theorem 2.2.2.10 (cf. [ASS06, Corollary I.6.10]). Let E be a finite-dimensional F-algebra. Then E contains a basic F-subalgebra Eb := eEEeE as above, and the naturalrestriction functor

res : RepE(F) −→ RepEb(F)

V 7→ eEV

is an equivalence of abelian categories with quasi-inverse −⊗Eb EeE.

Now let us observe that this equivalence extends from representations with coefficients Fto functors of families of representations over SpecF-schemes. Fix a choice of idempotents toproduce ι : Eb → E as above. We observe that the restriction functor res extends naturallyto families of representations over F-schemes X, via the natural transformations

res : RepE −→ RepEb ,

V 7→ eEV,

ρ : E ⊗F OX → EndOX (V ) 7→ (x 7→ ρ(eE)xρ(eE)) ρ (ι⊗OX).

of SpecF-scheme functors with a quasi-inverse as in the theorem above, so that it is anisomorphism. The induced isomorphic functor res : RepE → RepEb is given by sendingρ : E ⊗F OX → E to

(2.2.2.11) res(ρ) : Eb ⊗F OX −→ eEEeE.Now that we have given the association of representations, we are able to calculate the

line bundle Oθ(1) on RepE,D in terms of data native to the moduli problem for RepE,D wesummarize this calculation and the choices involved in this

Proposition 2.2.2.12. Let E be a F-algebra. Fix a choice of

(1) a set of primitive idempotents ejv indexed by a finite set Q0, v ∈ Q0, representingthe isomorphism classes of Definition 2.2.2.8, and the resulting idempotent eE =∑

Q0ejv and basic subalgebra Eb = eEEeE ⊆ E,

(2) a F-basis for ejv(J(Eb)/J(Eb)2)ejv , which produces the F-algebra homomorphismFQEb Eb → E of Theorem 2.2.2.6,

(3) a Cayley-Hamilton pseudorepresentation D : E → F, the associated semisimplerepresentation ρss

Dof E, the semisimple representation eEρ

ssD

of Eb, and the resultingdimension vector β for Q corresponding to the induced semisimple representation ofFQ via FQ Eb.

These choices define morphisms

RepE,D∼−→ RepEb,D → RepFQE ,D −→ RepQ,β.

Let θ : K0(E) −→ Z. Under these maps with the choices above, the line bundle Oθ(1) pullsback to

(2.2.2.13)⊗

v,w∈Q0

(∧βvβw (ejvEejw)

)n(v,w),

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where n(v, w) are a choice of integers as specified in (2.2.1.14).

Recall that Lemma 2.2.1.15 shows that there exist integers n(v, w) with the propertiesdemanded by (2.2.1.14).

Proof. On RepQ,β, we recall from (2.2.1.14) that we have a natural isomorphism

Oθ(1) ∼=⊗

v,w∈Q0

(∧βvβwHvw)⊗n(v,w).

We see in (2.2.2.7) that Hvw pulls back to RepF,Dβ as a direct summand of E ′, namely εvE ′εw,

where E ′ is the universal Azumaya algebra on RepFQ,Dβ . These idempotents εv ∈ FQE

correspond to the chosen idempotents ejvv∈Q0 of Eb; the homomorphism from FQE to E ′factors through Eb, so that the data of E ′ and εvEεw still make sense on the closed substackRepEb,Dβ ⊆ RepFQ,Dβ . Finally, if we write E for the universal Azumaya algebra on RepE,D,

we see in (2.2.2.11) that E ′ ' eEEeE. So the pullback of Hvw to RepE,D from RepQ,βD isexpressible in terms of its universal Azumaya algebra E as

ejveEEeEejw ∼= ejvEejw .

By combining King’s Theorem 2.2.1.12 with this calculation, we have the following de-duction. Recall the notation of θ-(semi)stability of a representation of E from Definition2.2.1.6.

Corollary 2.2.2.14. Let E be a F-algebra. Choose a character θ : K0(E) → Z and apseudorepresentation D : E → F with associated dimension vector βD ∈ ZK0(E) such that〈β, θ〉 = 0. If βD is indivisible and 〈β, θ〉 = 0, then the θ-stable locus of representations

of E descends to a quasi-projective subscheme Reps

E,D(θ) of RepE,D. If, moreover, βD is

stabilizing with respect to θ, then the Reps

E,D(θ) is a projective subscheme of the algebraic

stack RepE,D, with ample line bundle given in terms of the universal Azumaya algebra on

RepE,D by (2.2.2.13).

Note that while this subscheme is projective, it is not closed in RepE,D in any non-trivial case. The usual geometric situation is just like the standard construction of Pn asAn+1\0/Gm, lying inside [An+1/Gm].

Proof. By Theorem 2.2.1.12, the indivisibility of β implies that the θ-stable locusRep

s

E,D(θ) of the χθ-linearized GIT quotient of Rep,E,D

is a fine moduli space. As a result,

we have an immersion Reps

E,D(θ) → RepE,D. When βD is stabilizing with respect to θ, then

θ-semistability of representations of E is equivalent to θ-stability by Lemma 2.2.1.8. There-fore Rep

s

E,D(θ) is projective and a subscheme of RepE,D, and Proposition 2.2.2.12 identifies

a line bundle on RepE,D that is ample on Rep

s

E,D(θ).

2.2.3. Examples of Projective Moduli Spaces. We will give the motivating exam-ple, suggested by Kisin, of a moduli space of representations that is projective relative tothe moduli space of pseudorepresentations below. First we give an example of the projectivespaces constructed above.

Example 2.2.3.1. Let R be a finitely generated algebra over a field F. Choose n simplerepresentations ρ1, . . . , ρn of R of dimension di < ∞ over F. We stipulate that ρn 6' ρi for

98

1 ≤ i < n, but allow multiplicity among the ρi otherwise. Let ρ represent the d-dimensionaldirect sum

⊕n1 ρi. Let Dρ be the pseudorepresentation of R associated to ρ, i.e. Dρ := det ρ.

We will illustrate in this example that the moduli space of families of representations of Rwhose semisimplification is ρ and whose unique simple quotient is ρn is in fact projectiveover the point Dρ ∈ PsRd

R(F) with residue field FDρ .Let E be the universal Cayley-Hamilton representation of R over Dρ, i.e. E := R/CH(D).

By abuse of notation, we will write D for the factorization of D through E, and likewisefor the representations ρi. It is visible that Dρ is split over F. We know that E is finite-dimensional over F by Corollary 1.2.2.9, and we know from Theorem 1.4.3.1 that

RepR,D∼= RepE,D.

Consider now a character θ on K0(E) sending

θ :ρi 7→ 1 1 ≤ i < nρn 7→ −(n− 1)

Write β = βρ for the dimension vector of ρ, which is essentially the image of ρ in K0(E).

Now let us consider the projective SpecFDρ-scheme Repss

E,D(θ). We want to show that theconditions of Corollary 2.2.2.14 are satisfied. Here are the conditions:

• Since θ · β = 0, i.e. θ(ρ) = 0, it is possible for this space to be non-empty (the firstcondition of semisimplicity in Definition 2.2.1.6 is satisfied)• β is indeed indivisible – this is guaranteed because ρn appears with multiplicity 1

as a factor of ρ.• It is also the case that β is stabilizing with respect to θ (see Definition 2.2.1.9).

Simply see Example 2.2.1.10.

With these conditions satisfied, Corollary 2.2.2.14 now tells us that

RepsE,D(θ)∼→ Rep

ss

E,D(θ)

is a fine moduli space for θ-semistable (equivalently, θ-stable) representations of E lying overD ∈ PsRd

E, i.e. it is naturally a subscheme of RepE,D. We also know from Corollary 2.2.2.14

that the restriction of ψ : Repd

E → PsRdE to RepssE,D(θ) is projective.

Finally, we give a translation of the last bullet point above: a representation M withdimension vector β is θ-semistable (equivalently, θ-stable) if and only if its unique simplequotient is ρn. For if there exists some other simple quotient of M , then there exists a propersubrepresentation M ′ of M with ρn as a Jordan-Holder factor, implying that θ(M ′) < 0 andthat M is not θ-semistable. Conversely, if M is not θ-semistable, there must exist somesubrepresentation M ′ ⊂ M such that θ(M ′) < 0, which implies that ρn is a Jordan-Holderfactor of M ′ and that M/M ′ (which must have some simple quotient) has a simple quotientnot isomorphic to ρn.

We were motivated to investigate these projective spaces of representations by an exampleand suggestion of Kisin [Kis09a, §3.2, esp. Remark 3.2.7]. Kisin gives his construction andsuggestion in the context of continuous representations of a profinite group, but we will seelater (e.g. Theorem 3.2.4.1) that the continuous representations of a profinite algebra overa fixed finite field-valued pseudorepresentation amounts to the representations of a certainfinite-dimensional algebra over the finite field. Therefore these constructions of projectivespaces apply to Kisin’s context. Then the next paragraph §2.2.4 shows that deformations ofthese projective spaces are projective, as he suggests.

99

Definition 2.2.3.2. Let E be a finite-dimensional algebra over F with pairwise non-isomorphic simple representations ρi, 1 ≤ i ≤ n, each of dimension di. Write ρ =

⊕n1 ρi and

let Dρ (resp. βρ) be the corresponding pseudorepresentation (resp. dimension vector). Let

Rep′Dρ ⊂ RepDρ be the full subgroupoid of families of representations E ⊗F OX → E which

locally in the Zariski topology are of the form

(2.2.3.3) ρ '

ρ1 ∗ · · · · · ·0 ρ2 ∗ · · ·0 0

. . . ∗0 0 0 ρn

with the additional condition that follows. When we write

0 = L0 ⊂ L1 ⊂ · · · ⊂ Ln = M

for the filtration where Li/Li−1 ' ρi, we stipulate that the extension class of M/Li−1 as anextension of M/Li by ρi is non-trivial.5

Remark 2.2.3.4. As Kisin notes, this condition guarantees that such representationshave no non-trivial automorphisms, making the isomorphism (2.2.3.3) unique. The unique-ness only holds once one considers representations as maps into Azumaya algebras (an objectof Rep) instead of vector bundles with an action (an object of Rep). In the latter case, thetrivial (scalar) automorphisms are taken into account.

We immediately observe that this groupoid is a subgroupoid of the projective FDρ-subscheme Rep

ss

Dρ(θ) of RepDρ described in Example 2.2.3.1 above, where θ is the character

of K0(RepE) with θ : ρi 7→ 1 for 1 ≤ i < n and θ(ρn) = −(n − 1). Indeed, ρn is the unique

simple quotient of any object of Rep′Dρ , and this condition defines Rep

ss

Dρ(θ) in RepDρ . We

claim that Rep′Dρ is a closed subscheme of Rep

ss

Dρ(θ), and is therefore projective over SpecFDρas well.

Proof. First we fix certain idempotents in E. We know from Lemma 1.2.3.1 that therepresentation ρ has kernel precisely the Jacobson radical J(E) of E, and draws a surjection

ρ : E n∏i=1

Mdi(F).

Let ei represent a (non-canonical) lift to E of the idempotent of E/J(E) corresponding tothe identity element of Mdi(F) via ρ (see Remark 2.2.2.9). One can quickly check that theyremain pairwise orthogonal.

Let Euniv be the universal Azumaya algebra over RepssDρ(θ), receiving the universal rep-

resentation ηuniv from E ⊗F ORepssDρ (θ). These idempotents ηuniv(ei) along with the standard

reduced trace on Euniv correspond to the additional structure of a generalized matrix algebraof type (d1, . . . , dn) on the Azumaya algebra Euniv; we will use the notation of Lemma 2.3.1.4describing generalized matrix algebras to offer additional clarity to the following calcula-tions without requiring any additional theory of generalized matrix algebras. We have an

5Actuallly, Kisin uses the dual condition that the extension Li of ρi by Li−1 is non-trivial.

100

isomorphism

Euniv ∼=

Md1(A1,1) Md1×d2(A1,2) · · · Md1×dn(A1,n)Md2×d1(A2,1) Md2(A2,2) · · · Md2×dn(A2,n)

......

......

Mdn×d1(An,1) Md2(An,2) · · · Mdn(An,n)

,

where the Aij are line bundles on RepssDρ(θ) (with a canonical trivialization for Aii for each

i) and the algebra structure is determined by canonical isomorphisms

Mdidj(Aij)∼→ eiRej.

Consider a representation (η : E ⊗F OX → E) ∈ ob Repss

Dρ(θ), so that η = ηuniv ⊗RepssDρ

(θ)

OX . It inherits the structure of a generalized matrix algebra from Euniv, which we denoteagain with OX-line bundles Aij, abusing notation. The condition that η belongs to the

subgroupoid Rep′Dρ is equivalent to the triviality of the projection of the image of E in E to

Mdidj(Aij) viax 7→ η(ei) · x · η(ej)

for all pairs (i, j) such that 1 ≤ j < i ≤ n. To illustrate this equivalence, notice that thecondition for the pair (n, n− 1) is equivalent to the condition (in the language of Definition2.2.3.2) that the extension M/Ln−2 of M/Ln−1 by ρn−1 is non-trivial; following this, thecondition that the extension M/Ln−3 of M/Ln−2 by ρn−2 is non-trivial is expressed by thepairs (n, n− 1), (n, n− 2), (n− 1, n− 2); and so forth.

Notice in particular that the condition for pairs (n, j), 1 ≤ j < n, is what defines thesubscheme Rep

ss

Dρ(θ) ⊂ RepDρ ; this shows that Rep′Dρ is contained in the θ-semistable locus.It remains to show that the locus cut out by the condition that

η(eiEej) ⊂Mdidj(Aij) is trivial for 1 ≤ j < i ≤ n.

Let Nij ⊂ Mdidj(Aij) be the OX-submodule of Mdidj(Aij) generated by η(eixej) for x ∈E. This submodule will be trivial over the locus of support for the quotient moduleMdidj(Aij)/Nij, which is a closed subscheme.

Now we can apply these constructions to the universal representation ηuniv over Repss

Dρ(θ).

The intersection of all of these support loci is therefore the closed subscheme Rep′Dρ ⊂

Repss

Dρ(θ). Consequently, Rep′Dρ is projective over SpecFDρ .

We summarize what we have shown in the following theorem, confirming Kisin’s expec-tation [Kis09a, Remark 3.2.7] that the space Rep′Dρ is projective.

Theorem 2.2.3.5. Let ρi, 1 ≤ i ≤ n be pairwise non-isomorphic simple representationsof E with sum ρ, and let θ be a character of K0(RepE) sending ρi, 1 ≤ i < n to 1 and ρnto −(n− 1) as in the example above. The subgroupoid Rep

′Dρ ⊂ RepDρ defined by the condi-

tion (2.2.3.3) is a closed sub-SpecFDρ-scheme of Repss

Dρ(θ), and is consequently a projective

subscheme of RepDρ with ample line bundle Oθ(1).

2.2.4. Deformation of Ample Line Bundles. We conclude our discussion of thefibers of ψ by giving conditions such that the projective subspaces Rep

s

E,D(θ) ⊂ RepE,D

identified in the previous paragraph are the special fiber of a projective morphism to a localneighborhood on the base moduli space of pseudorepresentations. The question of which

101

condition we must impose has a fairly clear answer in light of the calculation of the ampleline bundle in Proposition 2.2.2.12: the idempotents on the fiber must be locally liftable to aneighborhood. By Lemma 2.1.3.2, this is true precisely for henselian local rings. To deformthe projectivity condition, we require that A is complete.6

Theorem 2.2.4.1. Let A be a complete local Noetherian ring with residue field FA andmaximal ideal mA. Let R be an A-algebra that is finite as an A-module, and write E forR ⊗A FA. Let R be equipped with a d-dimensional Cayley-Hamilton pseudorepresentationD : R→ A such that its special fiber D : E → FA is split. For any indivisible character θ ofK0(RepE), the line bundle Oθ(1) on the special fiber Rep

E,D is the restriction to the special

fiber of a line bundle defined over all of RepR,D.

In particular, if D is stabilizing with respect to θ, the projective fine θ-stable moduli spaceRep

,sE,D(θ) of Corollary 2.2.2.14 is the special fiber of a projective subscheme Rep

s

R,D(θ) of

the moduli stack RepR,D arising as the algebraization of the completion of RepR,D along

Rep,sE,D(θ).

Proof. Firstly, we show that the ample line bundle Oθ(1) is a specialization of an ampleline bundle that exists on all of RepD. This follows directly from the fact that we can liftthe idempotents defining Oθ(1) according to (2.2.2.13) from E⊗A FA to E. We then use thesame formula.

Now we apply formal GAGA [Gro61b, Theorem 5.4.5] to draw the conclusion.

We thank Mark Kisin for comments leading to the following remark.

Remark 2.2.4.2. In fact, Theorem 2.2.4.1 can be extended to a henselian base. The linebundle ample on the particular subspace certainly exists. Then, in place of the completionof RepR,D along Rep,s

E,D(θ), one can consider the henselization along this subscheme.

This theorem is especially significant in the context of continuous representations andpseudorepresentations of a profinite algebra. Of course, it is necessary to show that we canreduce the topological profinite case to the non-topological case under a finiteness conditionΦD, which we do in Chapter 3 (see e.g. Theorem 3.2.4.1). Then, firstly, the moduli space ofpseudorepresentations of a profinite algebra is a disjoint union of formal spectral of completelocal rings (Corollary 3.1.6.13). This allows Theorem 2.2.4.1 to be applied over the wholemoduli space of pseudorepresentations! Each component Spf BD arises from the completelocal ring BD, which is Noetherian upon the finiteness assumption ΦD. These notions willbe defined and discussed in Chapter 3. For now we discuss the moduli of representations ofa Cayley-Hamilton BD-algebra (R,D) where R is finite as a BD-module.

This is the context in which Kisin proposed the projectivity of a moduli formal schemeof representations of a profinite group with residually constant, split, multiplicity free pseu-dorepresentation D, and a certain ordering of non-trivial residual extensions of the represen-tation given in Definition 2.2.3.2 [Kis09a, Remark 3.2.7]. We verified in Theorem 2.2.3.5 that

the special fiber of ψ in this space Rep′E,D is projective and is a closed subspace of the larger

projective subscheme Repss

E,D(θ) ⊂ RepE,D. The ample line bundle Oθ(1) on Repss

E,D(θ) is

therefore also ample on Rep′E,D. Since this line bundle deforms to RepR,D as discussed in

Theorem 2.2.4.1, formal GAGA implies that the formal completion of Rep′E,D in RepR,D

6See Remark 2.2.4.2.

102

is projective. In particular, it is algebraizable and is a projective SpecBD-subscheme of

RepR,D, which we denote by Rep′R,D. This completes the confirmation of Kisin’s suggestion

that the space of representations with reduction in Rep′E,D is projective. We summarize this

in the following

Corollary 2.2.4.3. Let ρssD

: E → Md(FA) be chosen as in Theorem 2.2.3.5, and

choose an ordering of its simple factors in order to define the subgroupoids Rep′R,D ⊂

RepR,D,Rep′E,D ⊂ RepE,D as above. Assume that the associated pseudorepresentation satis-

fies condition ΦD. The formal completion of RepR,D along the projective subscheme Rep′E,D

of the special fiber of ψ is projective over Spf BD with ample line bundle Oθ(1). Conse-

quently, this formal scheme is algebraizable with algebraization Rep′R,D, a projective SpecBD-

subscheme of RepR,D.

2.3. Multiplicity Free Pseudorepresentations

Chenevier showed that a Cayley-Hamilton algebra (R,D) over a henselian local ring Awhich is residually split and absolutely irreducible is a matrix algebra (Theorem 2.1.3.3).This corresponds to very tidy results in in the moduli theory of representations and pseu-dorepresentations that locally satisfy these conditions: the deformations of representationsand pseudorepresentations are equivalent (Corollary 2.1.3.4).

Our goal in this section is to generalize these results to the case that a pseudorepresen-tation is residually multiplicity free (see Definition 1.3.4.4). Here, the moduli of representa-tions and pseudorepresentations are no longer equivalent. For example, over a multiplicityfree geometric point D of PsRd

R, non-trivial extensions of the Jordan-Holder factors Mi ofρssD

may form positive dimensional families of representations lying over this single pseu-dorepresentation; for example, this often happens if there exists Mi,Mj, i 6= j, such thatdimFA Ext1

R(Mi,Mj) ≥ 2. What we want to show is that around multiplicity free points inPsRd

R, ψ (resp. ψ) is an adequate moduli space. This will mean that the multiplicity freelocus of pseudorepresentations is a universal scheme-theoretic quotient for representations ofR up to conjugation. This improves the results of GIT (Theorem 1.5.4.2), which only havefine enough resolution to give a satisfactory theory for geometric points.

The main tool to accomplish this will be a generalized matrix algebra. The key resultgeneralizing Theorem 2.1.3.3 is Theorem 2.3.1.2, which shows that a Cayley-Hamilton alge-bra (R,D) over a henselian local ring is a generalized matrix algebra. The linear structurewe can put on the moduli space of representations of a generalized matrix algebra, an “adap-tation” of its representations, will allow us to show that the invariant functions on the spaceof framed representations are exactly the coefficients of the universal pseudorepresentation,as desired (Theorem 2.3.3.7).

Remark 2.3.0.4. Currently, the notion of generalized matrix algebra that we use is meantto work with pseudocharacters as opposed to pseudorepresentations (see §1.1.12). Therefore,we state our main result here in the case that (2d)! is invertible in A (i.e. charFA > 2d),which is a sufficient condition for pseudocharacters to be equivalent to pseudorepresentationsaccording to Proposition 1.1.12.3(3). We expect to develop a theory of generalized matrixalgebras compatible with pseudorepresentations, which will allow us to remove the conditionon the characteristic (see Remark 2.3.3.6).

103

2.3.1. Generalized Matrix Algebras. We define generalized matrix algebras.

Definition 2.3.1.1 ([BC09, Definition 1.3.1]). Let A be a commutative ring and R anA-algebra. We say that R is a generalized matrix algebra (GMA) of type (d1, . . . , dr) if R isequipped with

(1) a family of orthogonal idempotents e1, . . . , er of sum 1(2) for each i, an A-algebra isomorphism ψi : eiRei →Mdi(A),

such that the trace map T : R→ A defined by

T (x) :=r∑i=1

Tr(ψi(eixei)),

satisfies T (xy) = T (yx). The “data of idempotents” of the GMA is E = ei, ψi.

Here is the main result making possible our use of generalized matrix algebras to studyψ : Rep → PsR: given a henselian local ring A and a Cayley-Hamilton algebra (R,D) overA, R must be a generalized matrix algebra.

Theorem 2.3.1.2 ([Che11, Theorem 2.22(ii)]). Let D : R → A be a Cayley-Hamiltonpseudorepresentation over a henselian local ring A. If D is split and multiplicity free, then(R, TD) is a generalized matrix algebra.

We use the GMA structure on (R, E) to establish notation for elements of R analogousto matrices with a single non-zero entry.

Definition 2.3.1.3. Let Ek,li ∈ eiRei be the unique element mapping under ψi to the

matrix in Mdi(A) with 1 in the (k, l)th entry and 0 elsewhere. Write Ei = E1,1i . For

1 ≤ i, j ≤ r set Ai,j := EiREj. For 1 ≤ i, j, k ≤ r we have Ai,jAj,k ⊂ Ai,k so that theproduct in R induces maps

ϕi,j,k : Ai,j ⊗A Aj,k → Ai,k.

Bellaıche and Chenevier use these elementary matrix-like elements to exhibit a matrix-like structure on any given GMA (R, E).

Lemma 2.3.1.4 (Bellaıche-Chenevier, §1.3.2). There is a canonical isomorphism of A-algebras

R ∼=

Md1(A1,1) Md1×d2(A1,2) · · · Md1×dr(A1,r)Md2×d1(A2,1) Md2(A2,2) · · · Md2×dr(A2,r)

......

......

Mdr×d1(Ar,1) Md2(Ar,2) · · · Mdr(Ar,r)

,

where the A-algebra structure is determined by canonical isomorphisms

Mdidj(Aij)∼→ eiRej.

In analogy to Definition 2.2.1.4, we take note of the conditions that the maps ϕi,j,k mustsatisfy (cf. [BC09, Lemma 1.3.5]) as a result of the construction above. Here we implicitly use

a canonical morphism for each i, Ai,i∼→ A, that arises from the trace T , i.e. Ai,i → R

T−→ Ais an isomorphism.

(UNIT) For all i, Ai,i ∼= A and for all i, j, ϕi,i,j : A⊗Ai,j → Ai,j (resp. ϕi,j,j : Ai,j⊗A→ Ai,j)is the A-module structure of Ai,j.

(ASSO) For all i, j, k, l, the two natural maps Ai,j ⊗Aj,k ⊗Ak,l → Ai,l coincide.

104

(COM) For all i, j and for all x ∈ Ai,j, y ∈ Aj,i, we have ϕi,j,i(x⊗ y) = ϕj,i,j(y ⊗ x).

Bellaıche and Chenevier note that specifying the data of A-modules Ai,j, 1 ≤ i, j ≤ r withmaps ϕi,j,k as above satisfying (UNIT), (ASSO), and (COM), then R := Mdi,dj(Ai,j) isuniquely a GMA of type (d1, . . . , dr) such that EiREj ∼= Ai,j for all i, j. This completes asatisfying structure theory for GMAs.

Remark 2.3.1.5. Compare these conditions (UNIT), (ASSO), (COM) with the groupoidof families of quiver representations in Definition 2.2.1.4.

2.3.2. Trace Representations and Adapted Representations. Now we define thenotion of an adapted representation of a GMA (R, E). Adapted representations have extralinear structure that makes their moduli easier to handle than the general moduli problemof representations.

Definition 2.3.2.1 ([BC09, Definition 1.3.6]). Let B be a commutative A-algebra andlet (R, E) be a generalized matrix A-algebra. A representation ρ : R → Md(B) is said tobe adapted to E if its restriction to the A-subalgebra

⊕ri=1 eiRei is the composite of the

representation⊕r

i=1 ψi by the natural “diagonal” map

Md1(A)⊕ · · · ⊕Mdr(A)→Md(B).

We define RepAd(R, E) to be the functor associating an A-algebra B to the set of adapted

representations of (R, E) over B.

We also give a definition of a trace representation. This is nothing more than the ana-logue, where pseudocharacters replace pseudorepresentations, of the functor of representa-tions lying over a given pseudorepresentation.

Definition 2.3.2.2 ([BC09, §1.3.3]). If R is an A-algebra equipped with a d-dimensionalpseudocharacter T : R → A and B is a commutative A-algebra, we will say that a map ofA-algebras ρ : R → Md(B) is a trace representation if Tr ρ(x) = T (x)1B for any x ∈ R.We write Rep

T for the functor of trace representations on AlgA.

Of course, this definition can be applied to Azumaya algebra valued representations aswell, to get a groupoid RepT analogous to the definition for pseudocharacters (Definition1.4.1.1). We will assume that (2d)! is a unit in A so that we can consider pseudocharactersand pseudorepresentations to be the same object (cf. Proposition 1.1.12.3).

The key result is that a trace representation can be made into an adapted representationafter base change and conjugation. This is the key result we require in order to compare themoduli problem for adapted representations with our usual moduli problem for representa-tions.

Lemma 2.3.2.3 ([BC09, Lemma 1.3.7]). Let B be a commutative A-algebra and ρ : R→Md(B) be a trace representation. Then there is a commutative ring C containing B and aP ∈ GLd(C) such that PρP−1 : R → Md(C) is adapted to E. Moreover, if every finite typeprojective B-module is free, then we can take C = B.

We omit the proof, since we will give a proof more precisely tailored to the situation weare required to address in order to prove Proposition 2.3.3.5 below.

Adapted representations have a very concrete moduli functor.

Proposition 2.3.2.4 ([BC09, Propositions 1.3.9, 1.3.13]). When (R, E) is a GMA overA, the functor Rep

Ad(R, E) : AlgA → Set associating a commutative A-algebra B to the set

105

of homomorphisms R→Md(B) adapted to E is representable by a faithful A-algebra Bu withan injective universal adapted homomorphism R →Md(B

u).

Proof. The proof shows that one can find a ring Bu with inclusions Ai,j → Bu (whereAi,j are from Lemma 2.3.1.4 above) such that the isomorphism in Lemma 2.3.1.4 is preciselythe injection required. Bu is constructed as a quotient of the symmetric power algebra on⊕i 6=jAi,j. For additional details, see [BC09].

It follows from the existence of the universal adapted representation that the trace func-tion on a GMA is Cayley-Hamilton (cf. [BC09, Corollary 1.3.16]), where Cayley-Hamiltonis defined for pseudocharacters in analogy with the definition for pseudorepresentations inDefinition 1.1.8.5 (see [BC09, §1.2.3]). We give this brief argument: the trace T of theGMA data (R, E) is equal to the composition of the trace function Tr on Md(B

u) with theuniversal adapted representation R → Md(B

u) given by Proposition 2.3.2.4. Since Tr isCayley-Hamilton and R→Md(B

u) is an algebra homomorphism, so is T Cayley-Hamilton.

2.3.3. Invariant Theory of Adapted Representations. In this paragraph, our goalis to naturally identify the GIT quotient of Rep

Ad(R, E) with the algebra of traces, whichis A. This will allow us to do for adapted representations what we have not yet done forgeneral representations: show that pseudorepresentations are an adequate moduli space forrepresentations. After completing this paragraph, we will use the comparison of adaptedrepresentations with trace representations to show that ψ (resp. ψ) is an adequate modulispace over the multiplicity free locus of PsRd

R.Let (R, E) be a d-dimensional generalized matrix algebra of type β = (d1, . . . , dr). We

set up the notation for the following group schemes; the group Z(β) is made to act naturallyon the affine scheme Rep

Ad(R, E).

Definition 2.3.3.1. In analogy with automorphism groups of quiver representations,define GL(β) := GLd1 × · · · ×GLdr as a subgroup

GLd1 × · · · ×GLdr ⊂ GLd,

compatible with the maps ψi : Mdi → Md of Definition 2.3.2.1. Let Z(β) denote the centerof GL(β). Likewise, let PGL(β) denote the quotient of GL(β)/∆ of GL(β) by the diagonallyembedded central 1-dimentional torus ∆ ∼= Gm, and let PZ := Z(β)/∆.

Because Z(β) commutes with ⊕iMdi , its adjoint action preserves the adaptation. There-fore we have a natural action of Z(β) on Rep

Ad(R, E), inducing a natural action of PZ(β).There is a natural map

(2.3.3.2) RepAd(R, E) → Rep,d

R

given by forgetting the adaptation data. The map ⊕iMdi →Md induces a canonical injectionZ(β) → GLd (resp. PZ(β) → PGLd). In this lemma, we record the fact that (2.3.3.2) isequivariant for the action of Z(β) (resp. PZ(β)). We also calculate the GIT quotient of theaction of Z(β) on Rep

Ad(R, E).

Lemma 2.3.3.3. Given (R, E) a GMA over A of type β. The map (2.3.3.2) is equivariantfor the action of Z(β) (resp. PZ(β)). The invariant regular functions on Rep

Ad(R, E) underthis action are precisely A ⊂ Bu.

Proof. The first claim can be checked by each of the embeddings of functors and groupsset up above. For the claim on the invariant functions, as we mentioned in the proof of

106

Proposition 2.3.2.4, Bu is generated over A by Ai,j for i 6= j. As Z(G) ' Grm acts on Bu

(observe the form of the matrices in Lemma 2.3.1.4), it acts on each of Aij (i 6= j) by a(distinct) non-trivial character, namely, through the roots of GLd. Since these modules Aijgenerate the coordinate ring Bu of Rep

Ad(R, E), we see that (Bu)Z(β) ∼= A, i.e.

RepAd(R, E)//Z(β) ∼= SpecA

as desired.

If as usual, we let (R, E) be a d-dimensional generalized matrix A-algebra of type β, thelemma above shows that we have a morphism of stacks

[RepAd(R, E)/Z(β)] −→ RepR,T

[RepAd(R, E)/PZ(β)] −→ RepR,T ,

(2.3.3.4)

because of the equivariance of the adaptation-forgetting map (2.3.3.2) with respect to theembedding Z(β) → GLd (resp. PZ(β) → PGLd).

Now we will show that (2.3.3.4) is an isomorphism when A is a henselian local ring! Todo this, we will find a quasi-inverse. We recall here that we are assuming that (2d)! is aunit in A, so that pseudorepresentations and pseudocharacters are identical by Proposition1.1.12.3, and we can apply our knowledge of pseudorepresentations to this problem.

Proposition 2.3.3.5. Let A be a henselian local ring and let (R,D) be a d-dimensionalCayley-Hamilton A-algebra, so that (R, E) is a generalized matrix A-algebra with trace func-tion T . Then the natural induced maps of SpecA-algebraic stacks (2.3.3.4) are isomorphisms.

Remark 2.3.3.6. We record an alternative notion of generalized matrix algebra, replac-ing the notion relative to pseudocharacters with one for pseudorepresentations. Using thenotation of Definitions 2.3.1.1 and 2.3.1.3, we replace the trace map T with a “determinantmap” D : R → A as follows: let the symmetric group Sd act on the complete set of dprimitive orthogonal idempotents Ej,j

i ∈ R.

D(r) :=∑σ∈Sd

sgn(σ)∏

1≤i≤r

∏1≤j≤di

Ej,ji rσ(Ej,j

i ).

This determinant map is compatible with tensor products, and defines a d-dimensional pseu-dorepresentation. We expect to extend the theory of generalized matrix algebras of [BC09]to this case. This would eliminate the complications with the characteristic of coefficientrings.

Proof. (Proposition 2.3.3.5) Let X be a SpecA-scheme. Choose (ρ, VX) ∈ RepT (X).The idempotents ei ∈ R break VX into a direct sum of projective sub-OX-modules Vi := eiVBof rank di,

VX ∼=r⊕i=1

Vi.

Each Vi receives an A-linear action of eiRei ⊂ R, and therefore a OX-linear action of eiRei⊗AOX . Using the GMA data ψi : eiRei

∼→ Mdi(A), we see that EndB(Vi) ∼= Mdi(OX). Thismeans that as a OX-module, Vi is isomorphic to a twist of a free rank di vector bundle Fiby a line bundle Li.

Let Gi := IsomOX (Li,OX) be the Gm-torsor over X corresponding to Li. Then G :=×ri=1Gi is naturally a Z(β)-torsor. Indeed, the base change of VX to G from X is a free

107

rank d OG-vector bundle with a canonical basis adapted to (R, E). This defines a mapG → Rep

Ad(R, E), equivariant for the action of Z(β). We have therefore established amorphism

RepT −→ [RepAd(R, E)/Z(β)].

We observe that this provides a quasi-inverse to (2.3.3.4).

We now replace pseudorepresentations with pseudocharacters, using the fact that theyare equivalent to each other; this is the case because we are assuming that (2d)! is invertiblein the base ring A (cf. Proposition 1.1.12.3).

We recall the notation of §1.5. S is an affine Noetherian scheme, and R is a quasi-coherentfinitely generated OS-algebra. The map ν : Rep,d

R //PGLd → PsRdR measures the difference

between the GIT quotient of the space of framed d-dimensional representations Rep,dR by

the action of PGLd and the moduli scheme PsRdR of d-dimensional pseudorepresentations.

We showed in Theorem 1.5.4.2 that ν is a finite universal homeomorphism, or “almostisomorphism.” We showed in Corollary 2.1.3.4 that ν is an isomorphism in the neighborhoodof points corresponding to absolutely irreducible representations of R. Now we will extendthis result, showing that ν is an isomorphism in the neighborhood of points correspondingto multiplicity free pseudorepresentations.

Theorem 2.3.3.7. Let A be a commutative Noetherian ring and let R be a finitely gen-erated A-algebra. There exists a Zariski open subscheme U ⊂ PsRd

R with the following twoproperties:

(1) the set U contains all points of residue characteristic greater than 2d correspondingto multiplicity free pseudorepresentations of R, and

(2) ν is an isomorphism onto U .

Proof. We will write X = Rep,dR //PGLd for convenience.

We already know that ν is a finite universal homeomorphism of finite type SpecA-schemes(Theorem 1.5.4.2 ). Therefore ν is etale in the neighborhood of some point D ∈ PsRd

R ifand only if it is an isomorphism in that neighborhood.7 Since being an isomorphism is alocal property on the base, in order to prove the theorem, it will suffice to show that νis etale in a neighborhood of each of the specified points. Since ν is finite type, it willsuffice to show that the induced maps on complete local rings are etale; we will simply showthat they are isomorphisms. We may have to make an etale base change in order that thepseudorepresentation may be assumed to be split; this is not a problem, since we can descendthe etale property along this morphism.

We apply Theorem 1.4.3.1 to replace Rep,dR (resp. RepdR, resp. Rep

d

R) with Rep,dE,Du

(resp. RepdE,Du , resp. Repd

E,Du) where E = E(R, d). We will think of ψ (resp. ψ) as a

morphism out of RepdE,Du (resp. Repd

E,Du).

Let D be a point of PsRdR of residue characteristic greater than 2d, and write x =

ν−1(D) ∈ X for the corresponding point of X. We have a canonical map

OPsRdR,D−→ OX,x

7In fact, any etale universal homeomorphism is an isomorphism.

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which we wish to show is an isomorphism. Write UD := Spec OPsRdR,D, Vx := Spec OX,x. Of

course, UD classifies the pseudodeformations of D to Artinian A-algebras with residue fieldFD and has a universal pseudodeformation Du

D, so we will just write PsRDu

Din place of UD.

Because X and PsRdR are Noetherian, the morphisms PsRDu

D→ PsRd

R, Vx → X are flat.By the Artin-Rees theorem and the finitude of ν, they form a cartesian square

(2.3.3.8) Vx //

X

ν

ˆPsRDu

D

// PsRdR.

Now [Alp10, Proposition 5.2.9(1)] says that the flatness of the completion maps along withthe fact that the maps φ, φ of (1.5.2.2) are adequate moduli spaces (Definition 1.5.1.1) willimply that the maps φ, φ of

RepdE,Du ×PsRdRPsRDu

D

∼ // RepdE,Du ×X Vx

ψ

φ

Vx

ν

PsRDuD

PsRDuD

Vx

ν

OO

Repd

E,Du ×PsRdRPsRDu

D

∼ // Repd

E,Du ×X Vx

ψ

RR

φ

OO

are also adequate moduli spaces. Since (2.3.3.8) is cartesian, we get an identical picture by

replacing RepdE,Du×X Vx with RepdE,Du×PsRdRPsRDu

D, as we have indicated with the horizontal

isomorphisms above.Write pD for the prime ideal of ΓdA(R)ab corresponding to D ∈ PsRd

R, and write BD forthe pD-adic completion of (ΓdA(R)ab)pD . Now, unraveling definitions for the fiber product

RepdE,Du ×PsRdRPsRDu

D, using Lemma 1.1.8.6, and noting that the pD-adic completion ED

of E ⊗ΓdA(R)ab (ΓdA(R)ab)pD is isomorphic to E ⊗ΓdA(R)ab BD, we see that the fiber product isisomorphic to

RepdED,(D

u⊗ΓBD),

the groupoid of representations of the Cayley-Hamilton BD-algebra (ED, Du⊗BD) compat-

ible with its pseudorepresentation. The universal pseudorepresentation also is compatiblewith these completions and base changes; we write Du

D: ED → BD in place of Du⊗BD. Of

course, the same things can be said with Rep in the place of Rep.Because BD is a henselian ring, we see that we are now in the situation of Proposition

2.3.3.5. Indeed, Theorem 2.3.1.2 implies that (ED, TD) is a generalized matrix algebra, where

we write TD for the trace function ΛDuD

1,BDassociated to Du

D. Then Proposition 2.3.3.5 gives

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us isomorphisms of algebraic stacks

[RepAd(ED, TD)/Z(β)]

∼−→ RepDuD,

[RepAd(ED, TD)/PZ(β)]

∼−→ RepDuD

Lemma 2.3.3.3 tells us that

Γ(O(RepAd(R, E)))Z(β) ∼= BD.

This means that ψ (resp. ψ) is an adequate moduli space with source [RepAd(R, E)/Z(β)]

(resp. [RepAd(R, E)/PZ(β)]), since this situation from GIT outlined in Example 1.5.1.3

is an example of an adequate moduli space. Theorem 1.5.1.4(5) implies that adequatemoduli spaces arising from a reductive group acting on an affine scheme have a unique base.Therefore ν induces an isomorphism Vx

∼→ PsRDuD

as desired.

Corollary 2.3.3.9. Over the base locus defined in Theorem 2.3.3.7, ψ (resp. ψ) is anadequate moduli space.

This means that the pseudorepresentation scheme consists precisely of the invariant func-tions of the framed moduli scheme under the action of conjugation. This sort of statementon invariants is made clearly in Lemma 2.3.3.3.

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CHAPTER 3

Representations and Pseudorepresentations of Profinite Algebras

In this chapter, we apply the results on moduli spaces of representations of representationsand pseudorepresentations to the study of the moduli theory of continuous representationsof profinite algebras R. Our approach is to develop the topological theory of pseudorep-resentations and prove (see e.g. Corollary 3.1.6.13) the representability of their moduli byformal schemes that are disjoint unions of formal spectra PsRD

∼= Spf BD of complete localdeformation rings BD of residual pseudorepresentations D (pseudodeformation rings). Up tothis point we will have been following Chenevier [Che11]. Then, we give conditions for theNoetherianness of BD, the most important being known as ΦD (Definition 3.1.5.1). Then westudy the moduli space of representations more simply by studying the connected componentover each pseudodeformation spectrum. However, we will hold short of developing moduliformal schemes/algebraic stacks of representations directly. Instead, upon the assumptionof ΦD, we show that when the moduli problem of continuous representations is finitely pre-sented over the moduli of pseudorepresentations. Then, the moduli formal scheme/stacks ofcontinuous representations on formal schemes arise, over Spf BD, as completions of a naturalalgebraic, finite type scheme/algebraic stack of representations.

We accomplish this by showing that under the condition ΦD, the universal Cayley-Hamilton representation E(R,Du

D) of R over the universal deformation Du

Dof D is finite as

a module over BD. Then we simply observe that over coefficient rings that are separatedcontinuous BD-algebras, all (non-topological) representations of ED lying over Du

D|E are

automatically mD-adically continuous. Now, any representation of E(R,DuD

) is continuous,and we can apply the theory of Chapters 1 and 2 directly to show that the functors ofcontinuous representations on formal schemes over Spf BD are not only representable byadic formal schemes, but are algebraizable over SpecBD.

Not only can we apply representability results from the previous chapters, but the resultsof Chapter 3 overviewed above will allow us to apply all of the results of Chapters 1 and 2to this profinite topological case (assuming condition ΦD). We present these conclusions inTheorem 3.2.5.1.

3.1. Pseudorepresentations of Profinite Algebras

In this section we introduce continuous pseudorepresentations, due to Chenevier [Che11],and recall some basic topological facts about profinite rings and group algebras of profinitegroups. We assume that all topologies are Hausdorff. We will focus on working with profiniterings, and then pro-discrete rings.

Let A be a commutative topological ring and let R be a topological (continuous) A-algebra. We establish notions of continuity for pseudorepresentations of R.

Definition 3.1.0.10. With A,R as above, a d-dimensional pseudorepresentation D :R→ A is said to be continuous provided that the following equivalent conditions hold.

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(1) for each n ≥ 1, α ∈ In, the functions D[α] : R → A of Definition 1.1.2.14 arecontinuous.

(2) the characteristic polynomial functions Λi = ΛDi : R→ A, 1 ≤ i ≤ d are continuous.

(3) For every commutative continuous A-algebra B, the function DB : R ⊗A B → B iscontinuous.

We will show that the notions of continuity in the definition are indeed equivalent.

Proof. The equivalence of (1) and (2) is immediate from Amitsur’s formula (Proposition1.1.9.11(2)).

Recalling that a pseudorepresentation D : R→ A consists of a function DB : R⊗AB → Bfor every commutative A-algebra B, let us verify that (2) implies (3). When B is anycontinuous topological A-algebra, this definition does indeed guarantee that each of theinduced homogenous functions

DB : R⊗A B → B

that make up the polynomial law are continuous. For we can write B as a continuous quotientof a polynomial algebra C, where C is given its natural topology as a free A-module. TheD[α] are coefficient functions of DC by definitiion, and the functions DB are the compositionof DC with the continuous quotient map from C to B.

Conversely, using the case that B is a polynomial algebra B = A[t1, . . . , tn], we see that(3) implies (1). A polynomial coefficient D[α] for α ∈ Idd is the composition of a continuousmap

Rn −→ R⊗A A[t1, . . . , tn]

(ri) 7→n∑i=1

riti

followed by DA[t1,...,tn], followed by the continuous function from A[t1, . . . , tn] to A given by

taking the αth coefficient. Therefore D[α] is continuous, as desired.

3.1.1. Pro-discrete Topological Notions. We will be interested exclusively in eitherdiscrete or pro-discrete topologies. We begin by recalling some basic notions on profinitetopologies on rings, with an eye toward group algebras of profinite groups. We note thatrings are unital and associative but not necessarily commutative unless stated.

Lemma 3.1.1.1. Let R be a topological ring. The following conditions on R are equivalent.

(1) R is a profinite ring.(2) R is Hausdorff and compact.(3) R is Hausdorff, compact, and totally disconnected.(4) R is compact and has a fundamental system of neighborhoods of zero consisting of

open ideals of R.(5) There is an inverse system of finite discrete rings with surjective maps such that R

is its limit.

Proof. This is [RZ10, Proposition 5.1.2].

We will often denote by I a general open ideal of R.When A is a profinite (e.g. finite) commutative ring, we will be interested in the studying

continuous representations and pseudorepresentations of a profinite group Γ with coefficients

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in A or in commutative A-algebras. The group algebra A[Γ] is clearly not a profinite A-algebra. Therefore we discuss its natural topology and its profinite completion.

The topology on A[Γ] is defined by the fundamental system of neighborhoods of zerogiven by the kernels of the canonical surjections

(3.1.1.2) κ(I, U) := ker(A[Γ] −→ (A/I)[Γ/U ])

where I varies over open ideals of A and U varies over open normal subgroups of Γ. Each ofthese ideals have finite index in A[Γ]. We then define the complete group algebra to be the

completion of A[Γ] with respect to this topology,

A[[Γ]] := lim←−(A/I)[Γ/U ].

We see that this is a profinite ring, with open ideals

ker(A[[Γ]] −→ (A/I)[Γ/U ]),

where we abuse notation by writing κ(I, U) for these ideals of A[[Γ]] as well. It is also possibleto express the complete group algebra as the limit

A[[Γ]] ∼= lim←−U

A[Γ/U ].

Here are some basic facts about this construction.

Lemma 3.1.1.3. Let A be a commutative profinite ring and let Γ be a profinite group.

(1) The intersection of all the ideals of the form (3.1.1.2) is zero.

(2) A[Γ] is densely embedded in A[[Γ]].

(3) Γ 7→ A[[Γ]] behaves functorially in Γ.

Proof. This is [RZ10, Lemma 5.3.5].

One more notion that remains before discussing continuous pseudorepresentations ofprofinite algebras is that of the topology on the tensor product of pro-discrete algebras.Let us therefore be explicit in explaining this topology on these tensor products and theircompletions in the primary setting that we will require.

Definition 3.1.1.4. Let A be a profinite commutative ring. Let R be a profinite contin-uous A-algebra with a fundamental system of finite index ideals (Iλ). Let B be a continuouslinearly topologized commutative A-algebra1 with fundamental system of ideals (Jη). Thena neighborhood of ideals of 0 in R⊗A B is given by the ideals

Image(Iλ ⊗A Jη −→ R⊗A B)

as Iλ, Jη vary over elements of the fundamental systems of ideals mentioned above. Thecompleted tensor product is the limit

R⊗AB := lim←−λ,η

R/Iλ ⊗A B/Jη.

We observe that R⊗AB is profinite when R and B are profinite. The completed tensorproduct is, of course, complete, even if B is not complete with respect to its topology. Also,the natural map B → R⊗AB factors through the completion B. See [RZ10, §5.5] for somefurther discussion in the profinite case.

1See [Gro60, 0I, §7.1] for this definition.

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Remark 3.1.1.5. As discussed in Definition 3.1.0.10 and the proof the equivalence of thedefinitions of continuity given theret, a continuous pseudorepresentation consists of continu-ous functions DB : R⊗A B → B for every A-algebra B. Because all of the topological ringsinvolved are Hausdorff and the targets are complete, DB will factor uniquely through thecompletion map R ⊗A B → R⊗AB. When we need to distinguish these two cases, we willwrite DB for the map out of R⊗A B and DB for the map out of R⊗AB.

Now we would like to discuss continuous pseudorepresentations over profinite algebras.First let us specify the data that we start with.

Conventions. Our general setup is the following: A is a commutative profinite ring,and R is a profinite continuous A-algebra. It is important to note that if (Ii) are a set ofideals of A forming a fundamental system of neighborhoods around 0 in A, then the induced(Ii)-adic topology on R is not necessarily equivalent to the profinite topology on R. We willalways use the native profinite topology on R unless otherwise noted.

We are interested in continuous representations of R. We will generally use B to representa topological A-algebra of coefficients for the representation, or X for a Spf(A)-formal schemeof coefficients. For most of our discussion, we will let B be an admissible A-algebra,2 wherewe write AdmA for the category of admissible A-algebras. Sometimes B will be restricted tocertain subcategories of admissible A-algebras, such as local Artinian A-algebra with a fixedresidue A-field.

Any commutative profinite ring A is canonically a continuous Z := lim←−n Z/nZ-algebra.

Since Z ∼=∏

p Zp and the functors of representations over this base ring respect this de-composition, we will assume that A is a continuous Zp-algebra for some rational prime p.This means that p will be topologically nilpotent in the rings and algebras that we will beconcerned with.

3.1.2. Continuous Pseudorepresentations of Profinite Algebras. In this para-graph, we provide more characterizations of continuous pseudorepresentations in the caseof profinite or prodiscrete topologies, and show that the Cayley-Hamilton ideal CH(D) of acontinuous pseudorepresentation is closed.

The following lemma, due to Chenevier, shows that the conventional notion that a ho-momorphism from a profinite object to a discrete object is continuous if and only if it hasopen kernel extends to the case of pseudorepresentations.

Lemma 3.1.2.1 (Following [Che11, Lemma 2.33]). Let A be a profinite commutative ringand let R be a profinite A-algebra. Let B be a commutative continuous discrete A-algebra,and choose a B-valued d-dimensional pseudorepresentation D of R. Let PD denote thecorresponding degree d homogenous multiplicative A-polynomial law PD ∈Md

A(R,B). Thenthe following conditions are equivalent.

(1) D is continuous.(2) PD is continuous.(3) ker(D) is open.(4) ker(PD) is open.(5) D factors through a continuous discrete quotient ring of R⊗AB.(6) PD factors through a continuous finite quotient ring of R.

2See [Gro60, 0I, §7] for this and other notions for the topological coefficient rings and formal schemes.

114

Proof. The equivalences (3) ⇐⇒ (5), (4) ⇐⇒ (6) are clear.If ker(PD) (resp. ker(D)) contains an open ideal, then it is continuous because the char-

acteristic polynomial coefficient functions factor through a discrete space and have target adiscrete space. Therefore (3) =⇒ (1), (4) =⇒ (2).

Assume that PD is continuous; this means that each characteristic polynomial functionΛi : R → B is continuous. Since B is discrete and the topology on R is given by finiteindex ideals, each Λi factors through R/Ii for some open ideal Ii. The intersection of theseideals is open, so PD factors continuously through a finite discrete quotient. We have shown(2) =⇒ (4). The proof (1) =⇒ (3) is identical, except that the ideals topologizing R⊗ABare not necessarily finite index.

Since PD factors through D along the natural continuous map R → R⊗AB, we have(1) =⇒ (2). Now assume (3); we will prove (4). Since the contraction of ker(D) along thiscontinuous map is an open (equivalently, finite index) ideal and contained in ker(PD), we seethat ker(PD) is also finite index and therefore open.

We prove this lemma in a more generality than the profinite case, although we only provethe converse statement in the pro-discrete case.

Lemma 3.1.2.2. Let D : R → A be a d-dimensional pseudorepresentation between topo-logical rings. If D is continuous, then ker(D) ⊂ R is closed. If A,R are profinite as discussedabove and B ∈ AdmA, then the converse is also true.

Proof. The closure ker(D) of the two-sided ideal ker(D) is a two-sided ideal. Becausethe characteristic polynomial functions Λi : R → A are constant on cosets of ker(D) in Rand are also continuous, they are also constant on the closure of cosets. This means that Dfactors through R R/ker(D). According to Lemma 1.1.6.6, ker(D) is the largest two-sided

ideal K ⊂ R such that D factors through R/K. Therefore ker(D) = ker(D).Now we prove the converse statement, assuming that A,R are profinite and B ∈ AdmA.

Present B as a limit of A-algebras B = lim←−λBλ, where the system (Bλ) is composed offinite discrete continuous A-algebras and the maps are surjective continuous A-algebra ho-momorphisms. For each map πλ : B Bλ, let Pλ denote the induced polynomial lawDλ := πλ D. Lemma 3.1.2.1 tells us that ker(Dλ) ⊂ R ⊗A B is open and closed, andtherefore ker(D) =

⋂λ ker(Dλ) is closed.

In the non-topological case discussed in Chapters 1 and 2, the notion of a Cayley-Hamilton pseudorepresentation D : R→ A and a Cayley-Hamilton A-algebra (R,D) playeda large role. This will be especially true as we consider the moduli of representations ofprofinite algebras. Therefore the following lemma will be useful, showing that in the caseof profinite coefficients, the Cayley-Hamilton ideal CH(D) is closed, so that the naturalsurjection R R/CH(D) is continuous.

Lemma 3.1.2.3. Let A be a complete Noetherian local ring, with finite residue field FA.Let R be a profinite continuous A-algebra. Then CH(D) is a closed ideal. Consequently,R/CH(D) is profinite, the natural map R R/CH(D) is continuous, and is a continuousA-algebra.

Proof. Freely using the notation of Definition 1.1.8.5, we recall that CH(D) is the two-sided ideal of R generated by the image of χ[α](r1, . . . , rd) where α varies over Idd and ri vary

over R. Let RIdd have its standard set-theoretic meaning, i.e. the set of tuples of elements of

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R, each one corresponding to an element of Idd . Let Rl, Rr be copies of R distinguished for

notational purposes, and let (rαr ) denote an element of RIddl , and (ri) denotes an element of

Rd. Now define a function

RIddl ×R

d ×RIddr −→ R

((rαl ), (ri), (rαr )) 7→

∑α∈Idd

rαl · χ[α]D ((ri)) · rαr .

The image of this map is precisely the two-sided ideal generated by the image of the χ[α]D ,

i.e. CH(D).Because R is profinite, it is compact Hausdorff. And every map in sight is continuous.

Therefore the image CH(D) of the map above is closed by the closed map lemma.

3.1.3. The Φp Finiteness Condition on Profinite Groups. When we consider the

case that R = A[[Γ]], we will often want to impose a condition on Γ weaker than topologicalfinite generation, but strong enough to imply that the various functors of representations arefinite in the appropriate manner (e.g. finite type or Noetherian). This is the Φp condition,developed by Mazur [Maz89].

Definition 3.1.3.1. Let Γ be a profinite group and let p be a prime number. We saythat Γ satisfies the Φp finiteness condition when one of the following equivalent conditions

holds, for every finite index (and therefore open) subgroup H ⊂ Γ.

(1) The maximal pro-p quotient of H is topologically finitely generated.(2) For any finite dimensional Fp-vector space M with a continuous Fp-linear action of

H, the continuous cohomology group H1c (H,M) is finite dimensional over Fp.

(3) There are only a finite number of continuous homomorphisms from H to the additivegroup Fp.

Example 3.1.3.2. When K/Q` is a finite field extension, Γ = Gal(K/K) satisfies Φp

because Γ is topologically finitely generated.

Example 3.1.3.3. When F/Q is a finite field extension and S is a finite set of placesof F , let FS denote the maximal extension of F unramified outside S. Then by Hermite’stheorem, Gal(FS/F ) satisfies Φp.

Given a finite index subgroup H ⊂ Γ, there exists a maximal quotient Γ of Γ with theproperty that the image of H in Γ is pro-p. If Γ has property Φp, then one can check that

Γ (and of course the image of H in Γ) topologically is finitely generated. This quotient Γ is

called the p-completion of Γ relative to H. This notion will come up in the following sort ofexample.

Example 3.1.3.4 (cf. [Maz89, p. 389]). Let Γ satisfy condition Φp, and let F be a finite

characteristic p field. Fix a continuous homomorphism ρ : Γ→ GLd(F), where d ≥ 0. Thenfor any Artinian ring A with residue field F, the kernel of GLd(A) GLd(F) is a pro-p

group. Therefore the action of Γ through any deformation of ρ from F to A factors throughthe p-completion of Γ relative to ker(ρ).

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3.1.4. Continuous Deformations of a Finite Field-Valued Pseudorepresenta-tion. Let A be a Noetherian local commutative Zp-algebra with finite residue field FA, andlet R be a profinite A-algebra. Let F be a finite A-field (of characteristic p) and let

D : R⊗A F −→ Fbe a continuous d-dimensional determinant. We are interested in continuous deformationsof D.

It will be useful in the sequel to apply Theorem 1.3.1.1 and write ρssD

for a representative

ρssD : R⊗A F −→Md(F),

assuming that F is large enough so that D is split and therefore ρssD

exists over F.

Remark 3.1.4.1. Applying Theorem 1.3.1.3 along with the fact that finite fields areperfect, any finite field valued pseudorepresentation of R is automatically continuous, as isρssD

, since the kernel of such D must be finite index in R.

Artinian A-algebras with residue field F are the natural context to study deformationsof an object, such as D, defined over F. We reprise Definition 2.1.1.1.

Definition 3.1.4.2. Let AF be the category of Artinian local A-algebras with residuefield F, where morphisms are local and continuous A-algebra homomorphisms. For B ∈ AFwe write mB for its maximal ideal and endow it with the discrete (mB-adic) topology.

Let AF be the category of profinite local A-algebras with residue field F, where morphismsare local continuous A-algebra homomorphisms. For B ∈ AF we write mB for its maximalideal and endow it with the mB-adic topology.

The category AF includes AF as a full subcategory, and objects in AF consist of limits(filtered projective limits with surjective maps) in AF.

We define the deformation functor PsRD as follows.

Definition 3.1.4.3. With the data p,A,R, D, d and F as above, let PsRD be the co-variant functor on AF associating to each B ∈ ob AF the set of continuous d-dimensionalpseudorepresentations

D : R⊗AB −→ B

such that D⊗BF −→ F ∼= D. We call such deformations of D pseudodeformations.

Remark 3.1.4.4. Let us clarify the notation D⊗AB where D : R → A is a continuouspseudorepresentation from a profinite A-algebra R to a profinite commutative continuousA-algebra B. Let D ⊗A B : R⊗A B → B be the non-topological version of the base changeof D from A to B. The tensor product has a profinite topology defined by the ideals used inits profinite completion, although it is not complete with respect to this topology. Since thecharacteristic polynomial coefficient functions of D⊗A B are continuous and B is complete,they factor through the completion with respect to this topology – a full argument alongthese lines (but addressing a slightly different question) may be found in the proof of Lemma3.1.6.4. We denote this pseudorepresentation by D⊗AB. We will extend this notion whenwe allow B to be an admissible A-algebra.

We need to show that this definition of a functor is indeed functorial in AF and respectssurjective projective limits. It will suffice to prove functorality on AF and that it respectssurjective projective limits. This is due to Chenevier.

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Lemma 3.1.4.5 ([Che11, Lemma 3.2]). The functor PsRdR on AF is compatible with sur-

jective projective limits.

Proof. For a morphism (B → B′) ∈ AF and D ∈ PsRD(B), we observe that D⊗BB′ ∈PsRD(B′).

If R is any A-algebra, the functor of degree d homogenous multiplicative (not necessarilycontinuous) A-polynomial lawsMd

A(R,−) from A-algebras to sets is representable (Theorem1.1.6.5), and therefore commutes with projective limits. For finite continuous A-algebrasBi, a function R → lim←−iBi is continuous if and only if R → Bi is continuous for everyi. Applying this to the characteristic polynomial coefficient functions and recalling thedefinition of continuity of a pseudorepresentation, we see that the same equivalence appliesto pseudorepresentations. This completes the lemma.

Now we will show that the functor of continuous pseudodeformations of D : R⊗A F→ Fis representable.

Theorem 3.1.4.6 ([Che11, Proposition 3.3]). The functor PsRD : AF −→ Sets is rep-resentable, i.e. there exists a profinite local A-algebra BD and a continuous d-dimensionalpseudorepresentation

DuD : R⊗ABD −→ BD

such that for any B ∈ AF and any D ∈ PsRD(B), there exists a unique AF-morphismBD → B such that Du

D⊗BDB ∼= D.

Proof. We will construct the representing algebra BD as the profinite completion of therepresenting object in the analogous non-topological case. By Theorem 1.1.6.5, there existsa universal degree d multiplicative homogenous A-polynomial law

Du : R −→ ΓdA(R)ab

inducing the universal d-dimensional pseudorepresentation of Theorem 1.1.7.4 upon applying⊗AΓdA(R)ab.

Let ψ : ΓdA(R)ab → F be the A-algebra homomorphism corresponding to D. Call an idealI ⊂ ΓdA(R)ab open if I ⊂ ker(ψ), ΓdA(R)ab/I is a finite local ring, and the induced degree dmultiplicative A-polynomial law (called DI)

Du ⊗ΓdA(R)ab ΓdA(R)ab/I = DI : R −→ ΓdA(R)ab/I

is continuous. We must check that these ideals define a topology on ΓdA(R)ab. We will callthis topology the D-adic topology on ΓdA(R)ab.

As a union of ideals of I, I ′ of this type is a union of translates of I ∩ I ′, it will suffice toshow that I ∩ I ′ is open. We consider the canonical A-homomorphism

(3.1.4.7) ΓdA(R)ab/(I ∩ I ′) −→ ΓdA(R)ab/I × ΓdA(R)ab/I ′.

It will suffice to show that this map is a homeomorphism onto its image for the D-adictopology. As this map is injective and induces the diagonal map F → F × F after takingthe quotient of each of these three rings by their maximal ideals, we see that the properties“finite” and “local” are preserved. It remains to show that the DJ for J = I, I ′, I ∩ I ′ aretopologically compatible with this map (we will specify what this means below).

Recall that the continuity of a multiplicative homogenous polynomial law is defined interms of its characteristic polynomial coefficients. Write Λi/I for the reduction modulo I ofthe universal characteristic polynomial functions Λi : R → ΓdA(R)ab. By assumption, Λi/I

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and Λi/I′ are continuous. Now consider the commutative diagram

R

Λi/I×Λi/I′

!!Λi/(I∩I′) &&

Λi // ΓdA(R)ab

**ΓdA(R)ab/(I ∩ I ′)

(3.1.4.7)// ΓdA(R)ab/I × ΓdA(R)ab/I ′

As (3.1.4.7) is a homeomorphism onto its image for the discrete topology, we have thecontinuity of Λi/(I ∩ I ′). The fact that (3.1.4.7) is a homeomorphism onto its image for thediscrete topology implies that a quotient of its image will induce a continuous polynomiallaw if and only if a quotient of Λi/(I ∩ I ′) will. Since, as we noted above, the same claimwill hold true for the properties “finite” and “local,” we have shown that (3.1.4.7) induces ahomeomorphism onto its image for the topology on ΓdA(R)ab defined above, as desired.

Define BD to be the completion of ΓdA(R)ab with respect to this topology. This is a profi-nite A-algebra, by definition of the topology. There is a universal continuous d-dimensionalpseudorepresentation which we will call Du

D,

DuD : R⊗ABD −→ BD,

which we obtain from Du by the canonical map from ΓdA(R)ab to its completion BD. Weverify that this is an object of PsRD(BD) by applying Lemma 3.1.4.5:

PsRD(BD) = lim←−I

PsRD(ΓdA(R)ab/I),

and DI for each D-adically open ideal I defines a projective system of continuous ΓdA(R)ab/I-valued homogenous degree d multiplicative A-polynomial laws whose limit is Du

D.

Having constructed DuD

, we now verify its universality. Let B ∈ AF and choose D ∈PsRD(B). By Theorem 1.1.6.5, there exists a unique A-algebra map f : ΓdA(R)ab → B suchthat D = f Du and f (mod mB) ∼= ψ. Therefore ker(f) ⊂ ker(ψ). Also, ΓdA(R)ab/ ker(f) ⊂B is finite local. Finally, the continuity of D implies that ker(f) is open in the D-adictopology on ΓdA(R)ab. Therefore we have the universality of (BD, D

uD

) as a functor on AF;

Lemma 3.1.4.5 implies that it is universal on AF.

Remark 3.1.4.8. In the case that the profinite A-algebra R arises as the complete groupring R = A[[Γ]] for some profinite group Γ, one can alternatively form BD by replacing R

with A[Γ] and completing ΓdA(R)ab with respect to the D-adic topology described above.

This is Chenevier’s approach [Che11, Proposition 3.3]. Because A[Γ] is dense in A[[Γ]], onecan check the functors and constructions amount to the same thing.

Remark 3.1.4.9. We note that just as ΓdA(R)ab is generated by ΛDu

i (r) for r ∈ R and

1 ≤ i ≤ d (this is the non-topological case), so is BD topologically generated by ΛDuD

i (r) forr ∈ R and 1 ≤ i ≤ d. This is a consequence of Amitsur’s formula (Proposition 1.1.9.11(2)).This statement remains true when the choice of r ∈ R is restricted to r in a dense subsetof R. For example, if Γ ⊂ Γ is a dense subgroup of Γ, then a pseudodeformation of aD : A[[Γ]]⊗AF→ F is determined by the characteristic polynomial coefficients of the universalpseudodeformation of D evaluated at the elements of Γ.

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Remark 3.1.4.10. Following on Corollary 2.1.3.4, the continuous deformations of an ab-solutely irreducible residual pseudorepresentation are equivalent to continuous deformationsof the associated absolutely irreducible representation. Compare [Nys96, Theorem 2] in thecase of pseudocharacters.

3.1.5. Finiteness Condition ΦD. Having defined the universal continuous pseudode-formation of a finite field-valued continuous pseudorepresentation of a profinite A-algebraR, we are interested in finiteness properties of this functor. The main finiteness property ofinterest for the complete local continuous A-algebra representing this deformation functor isthe Noetherian property.

We recall Lemma 2.1.1.5, which gives equivalent conditions under which a complete localring is Noetherian. As in Chapter 2, we will aim to show the finiteness of the tangent spacein order to show that the pseudodeformation ring is Noetherian. Our strategy is to show thatthe tangent space is finite-dimensional if one assumes ΦD We have already developed thenon-topological notion of tangent spaces to a field-valued pseudorepresentation in §2.1.2, andhave given criteria for the finiteness of the dimension for this tangent space in Proposition2.1.2.3. We will freely use the notation of §2.1.2, and aim to prove a topological version ofProposition 2.1.2.3. As in §2.1.2, these are results of Chenevier, which we extend to arbitrarycharacteristic.

If D0 denotes a d-dimensional (possibly non-continuous) pseudorepresentation D : R →A, we recall that TD0 denotes the non-topological tangent space at D0 ∈ PsRd

R(A). Assumingnow that D0 is continuous, denote by T cD0

the A-submodule of continuous lifts of D0. Wecan write this as a union of A-modules

T cD0=⋃I

T ID0,

where I varies in the set of all open two-sided ideals of R such that ker(D0) ⊇ I, and T ID0is

defined to be the liftings P such that ker(P ) ⊇ I.Now assume that A is a finite field F and replace D0 with a continuous d-dimensional

pseudorepresentation D : R → F. Then SD := R/ ker(D) is finite dimensional by Theorem1.3.1.3.

Definition 3.1.5.1. With D : R → F a continuous pseudorepresentation of a profiniteF-algebra R into a finite field F, we say that D satisfies condition ΦD or that ΦD holds whenthe set of continuous extensions Ext1

R(SD, SD)c is finite dimensional as a F-vector space.

The finiteness condition ΦD on continuous extension is the finiteness condition we requireto give a topological generalization of Proposition 2.1.2.3.

Proposition 3.1.5.2 (Following [Che11, Proposition 2.35]). Let R be a profinite F-algebra where F is a finite field. Let D : R→ F be a continuous d-dimensional pseudorepre-sentation satisfying condition ΦD. Then T c

Dis finite dimensional over F.

Proof. It will suffice to show that there exists a bound on the dimension of T ID

thatis independent of choice of finite index two-sided ideal I ⊂ R such that I ⊂ ker(D). Fixsuch an ideal I. Also choose N such that ker(D) ⊂ CH(D); such an N exists by Lemma1.2.3.1(4). By Lemma 2.1.2.2, we know that

T ID ⊂ PdF(R/(ker(D)2N + I),F).

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Therefore, it will suffice to show that the right hand side has F-dimension bounded indepen-dently of I.

Since I ⊆ ker(D), we have for each n ≥ 1 the natural surjection

(ker(D)/(ker(D)2 + I))⊗nF −→ (ker(D)n + I)/(ker(D)n+1 + I),

similar to the proof of Proposition 2.1.2.3. Because SD is finite dimensional, it will be enoughto show that the F-dimension of ker(D)/(ker(D)2 + I) is bounded independently of I.

Because SD is a semisimple F-algebra, it will suffice to show that

dimF(HomR(ker(D)/(ker(D)2 + I), SD))

is bounded independently of I. By noting that the action of R on ker(D)/(ker(D)2 + I)factors through R/I and applying Lemma 1.3.3.1, we have

HomR(ker(D)/(ker(D)2 + I), SD)∼−→ Ext1

R/I(SD, SD).

The right hand side is a sub-F-vector space of Ext1R(SD, SD)c, i.e. the action of R on the

extension is continuous, since the action of R on any of these extensions factors throughthe finite (cardinality) F-algebra R/I. Because of the assumption that the dimension ofExt1

R(SD, SD)c is constant (and clearly independent of I), we are done.

Now we are ready to give some sufficient conditions on the continuous d-dimensionalpseudorepresentation D : R ⊗A F → F to guarantee that the deformation functor is repre-sented by a Noetherian ring. We recall that with D as specified above, ρss

D: R⊗A F→Md(F)

denotes a semi-simple representation associated to D by Theorem 1.3.1.1, which is continu-ous because the continuity of D implies that ker(D) is closed (Lemma 3.1.2.2). We will alsouse SD to denote SD := (R ⊗A F)/ ker(D), which is finite dimensional over F by Theorem1.3.1.3.

Theorem 3.1.5.3 (Following [Che11, Proposition 3.7]). Let A be a Noetherian completelocal Zp-algebra and let R be a profinite continuous A-algebra. Let D : R⊗AF → F be acontinuous d-dimensional pseudorepresentation, where F is a finite continuous A-field. Thenthe complete local profinite continuous A-algebra BD is Noetherian if any of the followingconditions are true.

(1) R is topologically finitely generated as an A-algebra.(2) D satisfies condition ΦD.

(3) Γ is a profinite group, R = A[[Γ]], and the continuous cohomology H1c (Γ, ad(ρss

D)) is

finite dimensional over F.(4) Γ is a profinite group satisfying Mazur’s Φp-condition and R = A[[Γ]].

Proof. We will show that any of these conditions implies that the tangent space TD(Definition 2.1.2.1) to PsRd

R at D is finite dimensional over F. This tangent space is nat-urally dual to mD/(mA,m

2D

), which is therefore finite-dimensional. Since A is assumed tobe Noetherian, this finiteness in turn implies that mD/m

2D

is finite-dimensional. Therefore,PsRD is Noetherian by Lemma 2.1.1.5.

That condition (2) is sufficient to prove that BD is Noetherian is immediate from Propo-

sition 3.1.5.2. Condition (3) is the same condition as (2) in the case that R = A[[Γ]], aftertensoring by ⊗FF. Condition (4) is sufficient to imply condition (3), as discussed in Definition3.1.3.1.

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Assume condition (1). Let Rfg be a finitely generated dense sub-A-algebra of R. ThenPsRd

Rfg is a finite type (hence Noetherian) A-scheme by Theorem 1.1.10.15. Upon observingthat PsRD is the formal scheme arising from PsRd

Rfg by completion at the maximal ideal ofPsRd

Rfg corresponding to D, we are done.

3.1.6. Pseudorepresentations valued in Formal Schemes. So far we have dis-cussed pseudorepresentations of a profinite A-algebra R, where A is a complete local Noe-therian Zp-algebra with finite residue field F. We have found that the functor of deformationsof a given finite field valued pseudorepresentation is representable (Theorem 3.1.4.6), andhave given sufficient conditions for it to be Noetherian (Theorem 3.1.5.3). We have restrictedourselves to profinite coefficient rings, in particular Artinian local commutative rings withfinite residue field. However, in order to discuss algebraic families of representations of aprofinite algebra, we will need to consider coefficient rings that are not profinite. For exam-ple, a one-dimensional family of representations will be valued in a polynomial ring like Fp[t].Our goal in this paragraph is to investigate the families of continuous pseudorepresentationsthat arise in these larger coefficient rings.

Our main result, Theorem 3.1.6.11, tells us that the study above is sufficient: even onlarger appropriately topologized rings, the universal pseudorepresentations are valued in acomplete local profinite ring. The first task must be to specify what exactly these largercoefficient rings are.

EGA contains the basic facts and terminology to describe linearly topologized rings andformal schemes. We will now freely use these terms, providing some references as we go. Wewill introduce here, however, some terminology that we have not found universal agreementupon, but which is an important distinction for our purposes.

Definition 3.1.6.1. Let A be a commutative adic Noetherian ring with ideal of definitionI. Let B be a linearly topologized commutative ring which is a continuous A-algebra.

(1) If B is topologically isomorphic over A to an admissible completion of a finitelygenerated A-algebra, then we say that B is topologically finitely generated as anA-algebra.

(2) If B is topologically isomorphic over A to the I-adic completion of a finitely gener-ated A-algebra, then we say that B is formally finitely generated as an A-algebra.Equivalently, B is a (continuous) quotient of a restricted power series over A.

We use this terminology in consonance with terminology established in [Gro60, 0I, §7;1, §10]. We are allow following the definition of “topologically finitely generated” used in[Che11, §3.9]: a completion of a finite type algebra. In particular, here are the correspondingdefinitions in the category of formal schemes.

Definition 3.1.6.2 ([Gro60, §0I, §10.13]). Let Y be a locally Noetherian formal schemewith ideal of definition K. Let f : X→ Y be a morphism of formal schemes. Then if any ofthe following equivalent conditions are satisfied, we say that f is formally finite type.

(1) X is locally Noetherian, f is an adic morphism, and if we write J := f ∗(K)OX, thenthe morphism f0 : (X,OX/J )→ (Y,OY/K) induced by f is finite type.

(2) X is locally Noetherian and is the inductive adic limit Xn over the inductive limitYn := Spec(OY/Kn) such that f0 : X0 → Y0 is finite type.

(3) Every point of Y is continued in an open formal affine Noetherian subschemes V ⊂ Ysuch that f−1(V ) is a finite union of open formal affine Noetherian subschemes Ui,

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such that the adic Noetherian ring Γ(Ui,OY) is formally finitely generated overΓ(V,OY).

When the context is clear, we say that such a morphism of formal schemes is simply“finite type.”

With these definitions in place, we can now specify the category of topological rings onwhich we will define the functor of pseudorepresentations, and later the functor and groupoidsof representations. We write AdmA for the category of continuous admissible A-algebras.

Definition 3.1.6.3. Let A be a commutative local complete Noetherian Zp-algebra withfinite residue field F and its adic topology. Let R be a profinite continuous A-algebra, whichwe assume to be complete and separated as an A-module. Let PsRd

R denote the functor

PsRdR : AdmA −→ Sets

sending B to the set of continuous B-valued d-dimensional pseudorepresentations of R,

D : R⊗A B → B.

We will often use the equivalent formulation in terms of a continuous homogenous degreed multiplicative A-polynomial law, which we will denote by P = PD, i.e. PD : R → B suchthat the induced multiplicative polynomial law R⊗AB → B is equal to D. Let us confirmthat these notions are indeed equivalent in this topological setting. We will writeMd

A(R,B)c

for the set of continuous degree d homogenous multiplicative A-polynomial laws from R toB.

Lemma 3.1.6.4. With A,R,D, d as above and B ∈ AdmA, the natural association

PsRdR(B) −→Md

A(R,B)c

(D : R⊗A B → B) 7→ D (Rid⊗1−→ R⊗A B)

is a bijection.

Proof. Clearly the map exists. We will exhibit a two-sided inverse. For P ∈MdA(R,B)c,

we have by e.g. Corollary 1.1.3.10(1) an induced determinant

DP : R⊗A B → B.

such that DP (r⊗ b) = bd ·DP (r). The characteristic polynomial functions ΛPi,A : R→ B are

continuous by assumption. Recalling that the characteristic polynomial functions are in factpolynomial laws ΛP

i : R→ B, we take the function associated to the A-algebra B,

ΛPi,B : R⊗A B −→ B ⊗A B,

which we concatenate with the A-algebra structure map B⊗AB → B to get a characteristicpolynomial function. This function is continuous, and it is also identical to ΛDP

i,B . This showsthat DP is continuous.

The equivalence of Lemma 3.1.6.4 makes it clear that PsRdR is a covariant functor: for

a morphism (ι : B → B′) ∈ AdmA and P ∈ MdA(R,B)c, we have PsRd

R(ι)(P ) := ι P ∈PsRd

R(B′).

Remark 3.1.6.5. The equivalence P ↔ DP described in the lemma above shows that thedescription of D⊗AB in Remark 3.1.4.4 extends to the case that B an admissible A-algebra,and also for X a Spf(A)-formal scheme since pseudorepresentations on X will be defined as

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a Zariski sheaf of algebra homomorphisms. The lemma above shows that one can simplyreduce to the underlying multiplicative A-polynomial law out of R in order to test continuity.

Example 3.1.6.6. The main example of A,R that we will concern ourselves with is thecase that A = Zp and R = Zp[[Γ]], where Γ is a profinite group.

Lemma 3.1.6.7 (Following [Che11, Lemma 3.10]). Let B ∈ AdmA, and let D : R⊗AB →B be a continuous d-dimensional pseudorepresentation D ∈ PsRd

R(B). Denote by C ⊂ B theclosure of the sub-A-algebra generated by the characteristic polynomial coefficients ΛP

i (r) forr ∈ R of the associated continuous homogenous multiplicative A-polynomial law P : R→ B.

(1) C is an admissible profinite sub-A-algebra of B. In particular, it is a finite productof local A-algebras with finite residue field.

(2) Assume that ι : B → B′ is a morphism in AdmA and let D′ : R ⊗A B′ → B′

be the induced continuous d-dimensional pseudorepresentation. Let C ′ ⊂ B′ be thesub-A-algebra associated to B′ as above. Then ι induces a continuous surjectionC C ′.

Proof. Assume that B is discrete, so that admissibility means that mnA ·B = 0 for some

n ≥ 1. Let P : R → B be the associated continuous multiplicative degree d A-polynomiallaw associated to D as above. By Lemma 3.1.2.1, P factors through some finite index,i.e. open two-sided ideal I ⊂ R containing mn

A · R. In particular, we can consider P tobe a polynomial law over the finite cardinality ring A/mn

A. Now ΓdA/mnA(R/I)ab is a finite

cardinality commutative ring, and therefore so is the ring C of the statement of the lemma,since C is, by Amitsur’s relations (Proposition 1.1.9.11(2)), the image of the A-algebra map

ΓdA/mnA(R/I) −→ B

canonically associated to P by Theorem 1.1.6.5.Now we consider the general case. Since B is admissible as an A-algebra, there is a

topological A-algebra isomorphism B∼→ limBλ, where Bλ is a discrete A-algebra and the

maps of the limit have nilpotent kernel. Write πλ : B → Bλ for the natural projection. LetP : R → B denote the continuous homogenous degree d multiplicative A-polynomial lawassociated to D. Write Pλ for πλ P .

Let C ⊂ B be the sub-A-algebra defined in the statement of the lemma. By the discretecase above, the image Cλ ⊂ Bλ of C ⊂ B in Bλ is of finite cardinality, and therefore

C∼−→ lim←−Cλ

is a profinite admissible A-subalgebra. The Jacobson radical of a profinite admissible ringmust include all ideals of definition. Therefore C/J(C) is finite, and part (1) follows.

For part (2), we simply note that the ring C ⊂ B is simply the closure of the inducedcanonical map ΓdA(R)→ B; this is functorial for ι : B → B′.

With this lemma controlling the characteristic polynomial coefficients of pseudorepre-sentations of R in place, we are going to show that the functor PsRd

R(A) of all continuousd-dimensional pseudorepresentations of R into admissible A-algebras is represented by thedisjoint union of deformation functors of finite field valued pseudorepresentations. We nowestablish the notation necessary to describe this result.

Definition 3.1.6.8. Denote by PsRdR(FA) the set of closed points of Spec(ΓdA(R)ab) with

finite residue field. We denote the associated pseudorepresentation by D and the point ofPsRd

R by SpecFD.

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By Remark 3.1.4.1, such pseudorepresentations and their associated semisimple repre-sentations are automatically continuous.

In the case that C is a local ring instead of being merely semi-local, then we know thatthe B-valued continuous pseudorepresentation D : R⊗AB → B induces a C/mC-valuedpseudorepresentation. This pseudorepresentation corresponds via representability of the(non-topological) pseudorepresentation functor to the canonical surjective map

ΓdA(R)ab C0 C0/mC0∼= C/mC ,

so C/mC is canonically isomorphic to FD for some D ∈ PsRdR(FA).

Definition 3.1.6.9. Let A,R,B,D, and C ⊂ B be as in the statement of Lemma3.1.6.7. If C is local, and C/mC is canonically isomorphic to FD as ΓdA(R)ab-algebras as perthe discussion above, we call D residually constant, and say that it is residually equal to D.

Now we define subfunctors of PsRdR on AdmA which are residually constant.

Definition 3.1.6.10. Let D ∈ PsRdR(FA). Let PsRD be the subfunctor of PsRd

R onAdmA defined by the following relation. For each B ∈ AdmA, let PsRD(B) ⊂ PsRd

R(B) bethe subset of d-dimensional pseudorepresentations that are residually constant and residuallyequal to D.

Lemma 3.1.6.7(2) shows that PsRD is indeed (covariantly) functorial in morphisms (B →B′) ∈ AdmA.

We have now defined two functors which we call PsRD. We will temporarily distinguishthese functors in order to show that they correspond in a natural way. Write PsRAdmA

Dfor the

functor of residually constant pseudorepresentations of Definition 3.1.6.10. Write PsRAFDD

for the deformation functor of the residual pseudorepresentation D : R⊗A FD → FD definedin Definition 3.1.4.3.

Theorem 3.1.6.11 (Following [Che11, Proposition 3.13]). Let A be a complete Noetherianlocal Zp-algebra with finite residue field, and let R be a profinite continuous A-algebra. Let

D ∈ PsRd,AdmA

R (FA). Then PsRAdmA

Dis representable by a local admissible A-algebra BD ∈

obAdmA whose residue field is canonically isomorphic to FD. Moreover,

(1) The W (FD)-algebra BD representing PsRAFDD

is canonically topologically isomorphic

to BD.(2) If ΦD holds, then BD is topologically finite type over A and Noetherian, and therefore

topologically finite type over Zp as well.

Proof. Lemma 3.1.6.7 implies that for any B ∈ obAdmA and any (P : R → B) ∈PsRd,AdmA

R (B), P is the composite of a continuous multiplicative polynomial law P ′ : R→ C

with C → B, where C ∈ AdmA is semi-local. If P ∈ PsRAdmA

D(B), then by definition

of the subfunctor, C is canonically a complete local A-algebra with residue field canonicalA-isomorphic to FD, i.e. C is canonically an object of AFD . Consequently, P ′ is naturally an

element of PsRAFDD

(C).Now Theorem 3.1.4.6 gives rise to a canonical continuous A-algebra homomorphism

BD → C corresponding to P ′, and whose universal pseudodeformation of D induces P ′.Composing this map with C → B, we have the representability result, as well as the canon-ical isomorphism BD

∼→ BD.

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Part (1) following directly from the arguments above along with Theorem 3.1.5.3; (2)follows directly from definitions.

It remains to address the representability of PsRdR. This is best done over the category

FSA of Spf(A)-formal schemes.

Definition 3.1.6.12. Let PsRdR = PsRd,FSA

R denote the contravariant functor sending aSpf(A)-formal scheme X to the set of continuous d-dimensional pseudorepresentations

R⊗Spf A O(X)→ O(X).

Likewise, for any D ∈ PsRdR(FA), let PsRFSA

D(X) ⊂ PsRd,FSA

R (X) define a subfunctor cut out

by the condition on D ∈ PsRd,FSAR (X) that for any open affine U ⊂ X, the restriction of D

to PsRd,AdmA

R (Γ(OU)) belongs to PsRAdmA

D.

Clearly the restriction of PsRd,FSAR to (AdmA)op coincides with the opposite functor of

PsRd,AdmA

R . Note that the D ∈ PsRd,AdmA

R (B) belongs to PsRAdmA

D(B) if and only if, for any

affine covering Spf(B) =⋃i(Ui), the image Di ∈ PsRd,FSA

R (Vi) of D belongs to PsRFSAD

forall i; this follows directly from Lemma 3.1.6.7. Now the same statement can be made ofPsRFSA

D: its restriction to (AdmA)op coincides with the opposite functor of PsRAdmA

D.

Corollary 3.1.6.13 ([Che11, Corollary 3.14]). Assume that condition ΦD holds for all

D ∈ PsRdR(FA). Then PsRd,FSA

Dis representable by the formal scheme∐

D∈FA

Spf(BD).

In particular, the functor PsRdR of continuous d-dimensional pseudorepresentations is locally

Noetherian and semi-local with local Noetherian component decomposition

PsRdR∼=∐D∈FA

PsRD.

As a result of the Theorem and Corollary, we will not bother to distinguish between

PsRAFDD

, PsRAdmA

D, and PsRFSA

D, and will simply denote these by PsRD and make the source

of the functor clear. Generally, it will be the category of admissible continuous A-algebrasAdmA or the category of Spf(A)-formal schemes FSA. We will also denote the object ofAdmA representing PsRD by BD, or by Spf(BD) ∈ obFSA.

3.2. Moduli of Representations of a Profinite Algebra

In analogy to §1.4 in the non-profinite case, we will introduce moduli spaces of topologicalrepresentations of the profinite A-algebra R. While we could proceed along the same lines as§1.4, defining functors and groupoids of representations fibered over the category of Spf(A)-formal schemes, then proving representability, etc., we will follow a different strategy. Underthe assumption of ΦD, we will show that the universal Cayley-Hamilton representationE(R,Du

D) of R over the universal pseudodeformation Du

Dof D is finite as a module over BD

and that its native profinite topology is equivalent to its mD-adic topology. This will allow usto deduce that the natural functor of continuous representations of R with constant residualpseudorepresentation D over Spf(A)-formal schemes can be found as the mD-adic completion

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of a finite type SpecBD-scheme/algebraic stack of (non a priori continuous) representationsof E(R,Du

D).

Throughout this section, A represents a complete Noetherian local ring with finite residuefield FA and maximal ideal mA. We write R for a profinite continuous A-algebra, not neces-sarily commutative. Of course, the topology on R is not necessarily the mA-adic topology.

3.2.1. Groupoids of Representations. Here are the functors and groupoids of rep-resentations of R that we will study on the category of Spf(A)-formal schemes FSA.

Definition 3.2.1.1. Let A and R be as specified above, and let d be a positive integer.

(1) Define the functor Rep,dR on FSA by

X 7→ continuous OX-algebra homomorphisms R⊗A OX −→Md(X).(2) Define the groupoid RepdR, fibered over FSA, by

obRepdR(X) = V/X rank d vector bundle,

continuous OX-algebra homomorphism R⊗A OX −→ EndOX(V )

and morphisms being isomorphisms of this data.

(3) Define the groupoid Repd

R, fibered over FSA, by

obRepd

R(X) = E a rank d2 OX-Azumaya algebra, with a

continuous OX-algebra homomorphism R⊗A OX −→ Eand morphisms being isomorphisms of this data.

The basic initial observations regarding these groupoids and the natural maps to PsRdR

hold in direct analogy to the non-topological case discussed in §1.4, although we hold off ondiscussing representability of these groupoids until §3.2.4. Namely, there are canonical maps

(3.2.1.2) Rep,dR −→ RepdR −→ Rep

d

R

in direct analogy to (1.4.1.2). Following §1.4.2, the reduced norm on Azumaya algebras,which is etale locally the determinant of a matrix algebra, allows us to associate to anyobject of these groupoids a d-dimensional continuous pseudorepresentation. We write thesemaps as

ψ : Rep,dR −→ PsRd

R,

ψ : RepdR → PsRdR, ψ : Rep

d

R → PsRdR.

Indeed, a pseudorepresentation induced by a continuous representation of R is continuous(see Definition 3.1.0.10) because the characteristic polynomial coefficient functions Λi : E →B on an Azumaya B-algebra E are continuous. This shows that the maps ψ, ψ, ψ are welldefined.

Just like (1.4.2.2), the canonical maps above form a commutative diagram

(3.2.1.3) Rep,dR

(1.4.1.2)//

ψ

))

RepdR(1.4.1.2)

//

ψ

##

Repd

R

ψ

PsRdR

127

This allows us to consider the Spf(A)-formal groupoids of representations as PsRdR-formal

groupoids. Now we establish notation to decompose the representation groupoids into thefiber of ψ (resp. ψ, resp. ψ) over each component PsRD ⊂ PsRd

R, D ∈ PsRdR(FA). Indeed, an

object of any of the representation groupoids over B ∈ AdmA induces a map Spf(B)→ PsRdR

via the appropriate ψ-map, and the condition that this map correspond to a residuallyconstant pseudorepresentation will define a PsRd

R-sub-fibered-groupoid, since we observe thatthe residually constant condition is stable under pullbacks in the category of PsRd

R-formalschemes (cf. Corollary 3.1.6.13).

Definition 3.2.1.4. For any D ∈ PsRdR(FA), we write Rep

D(resp. RepD, resp. RepD)

for the fiber of ψ (resp. ψ, resp. ψ) over the component PsRD ⊂ PsRdR.

Our next goal is to show that RepD

is representable by a Spf(A)-formal scheme, and,moreover, that condition ΦD implies that Rep

Dis formally finite type over PsRD, i.e. that

RepD

is a formally finite type Spf(BD)-formal scheme. While this may be shown ratherdirectly, we will deduce it from the finiteness result of the next paragraph.

3.2.2. Finiteness Results. In this paragraph, our goal is to prove Proposition 3.2.2.1.This proposition gives us the module-finiteness of the universal Cayley-Hamilton algebraassociated to R, whose definition we will recall below. This module-finiteness is the keyresult we require to prove the algebraizability of the representation functors on AdmA.

Proposition 3.2.2.1. Let B be a admissible A-algebra and let D : R ⊗A B → B bea continuous d-dimensional residually constant pseudorepresentation D. Assume that Dsatisfies ΦD. Then

(1) The B-algebra (R⊗A B)/CH(D) is finitely presented as a B-module.(2) The native pro-discrete topology on (R⊗A B)/CH(D) given by open ideals is equiv-

alent to the topology induced by a fundamental system of ideals for B.

First we require some lemmas.

Lemma 3.2.2.2. Let F be a finite characteristic p field, let R be a profinite F-algebra,and let D : R→ F be a continuous d-dimensional pseudorepresentation satisfying ΦD. ThenR/CH(D) is finite dimensional as a F-vector space and, equivalently, CH(D) is open as atwo-sided ideal of R.

Proof. We first note that the equivalence of the conclusions is immediate from R havingthe profinite topology.

Replace R with R/CH(D), so that (R, D) is a Cayley-Hamilton F-algebra. Let S :=R/ ker(D), which we know from Theorem 1.3.1.3 to be a finite dimensional semisimple F-algebra. It is naturally a quotient of R by Lemma 1.2.1.1. The proof of Proposition 2.1.2.3shows that the assumption ΦD is sufficient to imply that R/ ker(D)n is finite dimensional asa F-vector space for any n ≥ 0. Now by Lemma 1.2.3.1, ker(D) is nilpotent. This completesthe proof that R is finite dimensional.

Remark 3.2.2.3. We emphasize that in the proof above, we do not assume that CH(D)is a closed ideal of R, nor, equivalently, that the natural surjection R/CH(D) is continuous.This fact is a consequence of the proof.

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Lemma 3.2.2.4 (Infinite Nakayama Lemma). Let A be a complete Noetherian local ringwith maximal ideal mA and residue field FA and let M be an A-module. Assume that M is mA-adically separated, i.e.

⋂i≥0 m

iA ·M = 0, and assume that M/mA ·M is finite dimensional as

a FA-vector space. Then M is a finite A-module generated by any set of lifts for a generatingset for M/mA ·M . In particular, one can apply the (standard) Nakayama Lemma to M .

Proof. Choose a basis m1, . . . , mn for M/mA ·M , and let m1, . . . ,mn be a choice oflifts to M for the basis. Choose 0 6= x ∈M , and let k ≥ 0 be the greatest integer such thatx ∈ mk

A ·M ; write xk for x. Because (mi) is a basis and the Noetherianness of A impliesthat mb

A/mb+1A is finite dimensional over FA for all b, there exists an A-linear combination∑n

1 aikmi such that aik ∈ mkA and

(3.2.2.5) xk −n∑i=1

aikmi ∈ mk+1A ·M

Now set xk+1 to this difference, and choose ai,(k+1) ∈ mk+1A , 1 ≤ i ≤ n such that (3.2.2.5)

is satisfied with k + 1 in place of k; iterate this process for all j ≥ k, generating xj, aij forj ≥ k, 1 ≤ i ≤ n.

Now set, for each i, 1 ≤ i ≤ n,

ai :=∞∑j=k

aij ∈ A,

where the sum is convergent because A is mA-adically complete and aij ∈ mjA for any j ≥ k.

Observe that

x−n∑i=1

aimi ∈ mjA ·M

for any j ≥ k. Therefore, by the separation hypothesis on M , x =∑n

1 aimi. This showsthat (mi) is an A-basis for M , as desired.

Now we can prove Proposition 3.2.2.1

Proof. First, we will prove the result when B is discrete. We already know that CH(D)is a two-sided ideal of R⊗A B, so we must show that it is open.

By Lemma 3.1.6.7 and the definition of residual constancy of D (Defintion 3.1.6.10), Dfactors through a finite cardinality Artinian local sub-A-algebra C ⊂ B (C is the image ofthe canonical continuous homomorphism BD → B) with residue field FD, i.e. there exists acontinuous deformation

DC : R⊗A C → C

of D inducing D upon ⊗CB.Consider the Cayley-Hamilton quotient (R⊗A C)/CH(DC). Using the canonical surjec-

tion C → C/mC∼→ FD, we tensor DC by ⊗CFD. Now Lemma 1.1.8.6 implies that we have

an isomorphism

(R⊗A C)/CH(DC)⊗C FD∼−→ (R⊗A FD)/CH(D).

Applying our assumption that ΦD holds, Lemma 3.2.2.2 tells us that the right hand side isa finite dimensional F-vector space. Since the C-algebra (R⊗A C)/CH(DC) is trivially mC-adically separated because C is Artinian, the “infinite Nakayama lemma” (Lemma 3.2.2.4)implies that it is also finite as a C-module. Since all of the involved rings are profinite and

129

the maps factor through profinite completions, we may apply Lemma 3.1.2.3 so that we knowthat the factor map

(R⊗A C) −→ (R⊗A C)/CH(DC)

is continuous and CH(DC) is closed in R⊗AC. The target is also finite cardinality, showingthat CH(DC) is also an open ideal of R⊗AC. We have now completed the proof in the casethat B is a finite Artinian ring.

Now we deduce the general discrete case over B from the local discrete case completedfor C above. The natural map

(R⊗A C)/CH(DC)⊗C B −→ (R⊗A B)/CH(D)

exists and is an isomorphism by Lemma 1.1.8.6; it is continuous and CH(D) ⊂ (R⊗A B) isopen, as the natural topology on both sides is discrete, proving part (2). Since the left handside is finitely presented as a B-module by the arguments above, so is the right hand sideproving part (1). This completes the argument.

Now we no longer assume that B is discrete. We may write B as a limit of discretecontinuous A-algebras B = lim←−λBλ where the maps are surjective with nilpotent kernel, andwrite Cλ ⊂ Bλ for the algebra C in the discrete case above. Then C = lim←−λCλ is a completelocal Noetherian sub-A-algebra of B with residue field FD, since we are assuming conditionΦD and may apply Theorem 3.1.5.3. Write πλ : B → Bλ for the natural surjections. WriteDλ for πλ D; it is a continuous d-dimensional pseudorepresentation

Dλ : R⊗A Bλ → Bλ

that satisfies the conditions of the discrete pseudorepresentation called “D” above. Likewise,we write DCλ the pseudorepresentations called “DC” above.

Consider the Cλ-algebra homomorphism

(R⊗A Cλ)/CH(DCλ) −→ (R⊗A Cλ′)/CH(DC′λ).

By Lemma 1.1.8.6, it becomes an isomorphism after applying ⊗CλCλ′ to the left side. There-fore, it is continuous and surjective; it has nilpotent kernel since the maps Cλ → Cλ′ do too.By the same reasoning, for every λ there is a canonical surjection

(R⊗A C)/CH(DC) −→ (R⊗A Cλ)/CH(DCλ).

Therefore, the image of the natural map

(3.2.2.6) (R⊗A C)/CH(DC) −→ (R⊗AC)/CH(DC)

is dense, since the right hand side surjects onto each (R⊗ACλ)/CH(DCλ) as well. The map

(3.2.2.6) is also injective, since CH(DC) is dense in CH(DC). We now aim to show that it isan isomorphism.

We know that CH(DC) ⊂ R⊗AC is closed by Lemma 3.1.2.3, so we have

(R⊗AC)/CH(DC)∼−→ lim←−

λ

(R⊗AC)/CH(DCλ πλ).

Now [Gro60, Proposition 10.10.5] and the finiteness of (R⊗AC)/CH(DCλ) as Cλ-modulesproved in the discrete case above, (R⊗AC)/CH(DC) is a finitely generated C-module. Al-ternatively, we can apply the infinite Nakayama lemma again. We note that the mC-adictopology on (R⊗AC)/CH(DC) is equivalent to its profinite topology arising from the com-plete tensor product.

130

Now we observe that (3.2.2.6) is an isomorphism of finite C-modules. Indeed, the imageis a dense sub-C-module of a finite C-module. We have now completed the proof in the casethat B was a Noetherian complete local ring.

We now deduce the general case from what we have done. Lemma 1.1.8.6 shows us thatwe have a natural isomorphism

(3.2.2.7) (R⊗A C)/CH(DC)⊗C B∼−→ (R⊗A B)/CH(D),

since D arises from DC by ⊗CB. We conclude that the right hand side is finitely presented asa B-module, since C is Noetherian, proving (1). The compatibility of (3.2.2.7) with ⊗BBλ,yielding the isomorphism of discrete algebras

(R⊗A Cλ)/CH(DCλ)⊗Cλ Bλ −→ (R⊗A Bλ)/CH(Dλ),

shows us that (2) is true.

3.2.3. Universality Results. Recall the (non-topological) notion of universal Cayley-Hamilton representation of R (§1.2.4). This is a ΓdA(R)ab-algebra

E(R, d) := (R⊗A ΓdA(R)ab)/CH(Du),

with the data of the universal pseudorepresentation Du |E: E(R, d) → ΓdA(R)ab and thecanonical quotient map from R⊗AΓdA(R)ab. We have shown in Theorem 1.4.3.1 that modulispaces of d-dimensional representations of R are equivalent to their counterpart modulispaces of d-dimensional representations of E(R, d). Our goal in this paragraph is to provethis result in the profinite topological setting of this chapter.

We will carry out this task over each component PsRD of PsRdR. There is no significant

loss of generality in doing this. Let us establish the notation for these universal Cayley-Hamilton algebras.

Definition 3.2.3.1. Let D ∈ PsRdR(FA). The universal Cayley-Hamilton representation

over PsRD, denoted E(R,DuD

), is the BD-algebra

E(R,DuD) := (R⊗A BD)/CH(Du

D),

often considered with its canonical factor map ρuD

: R ⊗A BD → E(R,DuD

). We establishnotation for the completed case as well,

E(R, DuD) := (R⊗ABD)/CH(Du

D),

with the canonical factor map ρuD

: R⊗ABD → E(R, DuD

).

Before proving the universality theorem for the Cayley-Hamilton algebra E(R,DuD

), wepoint out the consequences of ΦD for this algebra. This theorem follows directly fromProposition 3.2.2.1, and the last part from Corollary 1.2.2.10.

Theorem 3.2.3.2. Assume that D ∈ PsRdR(FA) satisfies ΦD. Then

(1) The natural profinite completion map

E(R,DuD) −→ E(R, Du

D)

is an isomorphism.(2) E(R,Du

D) is finite as a BD-module

(3) The native topology on E(DuD

) is equivalent to the its mD-adic topology as a BD-module.

(4) E(DuD

) is finite as a module over its center and is a Noetherian ring.

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Now we prove an analogous result in our profinite topological setting to the universalityof the Cayley-Hamilton algebra (Proposition 1.2.4.3) and the resulting equivalence of repre-sentation categories between R and E(R, d) (Theorem 1.4.3.1). The non-topological resultsproduce universal maps, and we check that they are continuous.

We require some notation. Following the convention that RepD

denotes the fiber of

Rep,dR over PsRD, denote by Rep

D|Ethe fiber of Rep,d

E(R,DuD

) over PsRD|E ⊂ PsRdE(R,Du

D).

Theorem 3.2.3.3. Let X be a Spf(A)-formal scheme. Any representation in the formalgroupoids Rep

D(X),RepD(X),RepD(X) factors uniquely continuously through the universal

Cayley-Hamilton representation ρuD⊗BD O(X). This factorization induces equivalences of

PsRD-formal groupoidsRep

D

∼−→ RepD|E ,

RepD∼−→ RepD|E ,

RepD∼−→ RepD|E .

Proof. It will suffice to work formally Zariski-locally on X, so we may replace OX withan admissible A-algebra B. As in the proof of Theorem 1.4.3.1, it will suffice to work witha continuous B-algebra homomorphism ρ : R⊗AB → E in RepD(B), since objects of theother groupoids amount to additional data on top of the rank d2 Azumaya B-algebra E andthe map ρ.

Recall Definition 1.2.4.1, which is the notion of a Cayley-Hamilton representation of R.Following Remark 1.2.4.2, we note that a the data of ρ induces a d-dimensional Cayley-Hamilton representation of R over B, namely

(B, (E , det), ρ),

where det : E → B represents the reduced norm map for the Azumaya B-algebra E .Proposition 1.2.4.3 shows that the universal d-dimensional Cayley-Hamilton represen-

tation (ΓdA(R)ab, (E(R, d), Du|E), ρu) is initial in the category CHd(R) of Cayley-Hamiltonrepresentations of R. Thus there exists a canonical CHd(R)-morphism

(ΓdA(R)ab, (E(R, d), Du|E), ρu) −→ (B, (E , det), ρ).

We know that the map ΓdA(R)ab → B included in this data is continuous with respect to thetopology on ΓdA(R)ab defined in Theorem 3.1.4.6 for the choice of D ∈ PsRd

R(FA), since B hasresidually constant pseudorepresentation D. The completion with respect to this topologyis BD and B is complete, so that we have a map BD → B factoring ΓdA(R)ab → B.

Therefore the continuous B-algebra homomorphism E(R, d)⊗ΓdA(R)abB → E which is part

of the data of the morphism in CHd(R) factors through E(R, d)⊗ΓdA(R)ab BD. Recalling that

E(R, d) := (R⊗A ΓdA(R)ab)/CH(Du), we have by Lemma 1.1.8.6 a canonical isomorphism

E(R, d)⊗ΓdA(R)ab BD∼−→ (R⊗A BD)/CH(Du

D) ∼= E(R,DuD),

so that we now have a canonical continuous map E(R,DuD

)⊗BD B → E factoring ρ.

We have now exhibited a PsRD-groupoid morphism RepD → Repd

E(R,DuD

). We observe

that this lies inRepD|E because ΓdA(R)ab → B factors through BD. The map ρuD

: R⊗ABD →E(R,Du

D) induces an inverse morphism by composition.

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Here is an interesting consequence of this universality. Once we show that the groupoidsare representable by formal algebraic stacks, this corollary says, essentially, that ψ, ψ, ψ areadic morphisms.

Corollary 3.2.3.4. As usual, let A be a commutative Noetherian local profinite ring, letR be a profinite A-algebra, and choose D ∈ PsRd

R(FA) satisfying ΦD. Choose an admissibleA-algebra B along with a continuous d-dimensional representation ρ : R ⊗A B → E ofresidually constant pseudorepresentation D. Then ρ is still continuous with respect to thefiner mD-adic topology on B.

Proof. Let B, ρ be as in the statement of the corollary. Theorem 3.2.3.3 implies thatwe have a continuous map BD → B and a canonical continuous factorization of ρ, and acontinuous BD-algebra map ED → E , through which R→ E factors. The fact that BD → Bis continuous means that the mD-adic topology on B is (not necessarily strictly) strongerthan its native topology. Clearly if we topologize E with respect to mD, the map ED → Ewill remain continuous.

3.2.4. Representability Results. Now we will work toward showing that the for-mal groupoids Rep

D, RepD, RepD are representable by PsRD = Spf(BD)-formal schemes.

In fact, we will show much more, using the universality (Theorem 3.2.3.3) and finiteness(Theorem 3.2.3.2) of the universal Cayley-Hamilton representation of R of residually con-stant pseudorepresentation D. We will show that the d-dimensional representation groupoid

Rep,dE(R,Du

D) (resp. RepdE(R,Du

D), resp. Rep

d

E(R,DuD

)) for E(R,DuD

) is formally finite type over

PsRD and will prove this by showing that it is algebraizable with finite type algebraization

Rep,dE(R,Du

D) (resp. RepdE(R,Du

D), resp. Rep

d

E(R,DuD

)). This will show, by Theorem 3.2.3.3, that

the representation groupoids above are topologically finite type, Noetherian formal schemesover Spf(A) that are formally finite type over PsRD = Spf(BD).

In order to prove algebraization of the formal PsRD-groupoids of representations ofE(R,Du

D), we need to find algebraic groupoids of continuous representations. In fact, what

we will show is that, after applying a natural topology to groupoids of non-topological rep-resentations such as RepdE(R,Du

D), this non-topological groupoid of representations consists

entirely of continuous representations. This result depends critically on the finiteness condi-tion ΦD and the work done in the previous paragraphs.

Theorem 3.2.4.1. With A,R, d, D, and E(R,DuD

) as above, assume that ΦD is true.Then the restrictions to admissible BD-algebras of the non-topological SpecBD-groupoids ofrepresentations Rep

E(R,DuD

),DuD

, RepE(R,DuD

),DuD

, and RepE(R,DuD

),DuD

of E(R,DuD

) lying over

PsRD are equivalent to their counterparts RepD

, RepD, and RepD.

Proof. Because ΦD is satisfied, Theorem 3.2.3.2 gives us that the BD-algebra E(R,DuD

)

is module-finite and its native topology as a quotient of R⊗ABD is identical to its mD-adictopology. Choose B ∈ AlgBD that is an admissible BD-algebra, and choose a non-topological

representation ρ ∈ Repd

E(R,DuD

),DuD

(B). This is the data

ρ : E(R,DuD)⊗BD B −→ E

where E is a rank d2 B-Azumaya algebra and det ρ = (BD → B) DuD

. Rembering thetopology on B, we have a topology on both the source and the target of ρ, and we observe that

133

ρ is a map of finitely presented B-modules, and is therefore continuous under the mD-adic

topology. Therefore ρ ∈ Repcont

E(R,DuD

),DuD

(B).

Likewise, one can start with ρ ∈ Repd

E(R,DuD

),DuD

(B), and observe that forgetting the

topology gives us an object of Repd

E(R,DuD

),DuD

(B), providing a quasi-inverse morphism.

Remark 3.2.4.2. Let us note what may go wrong when ΦD is not satisfied. The Cayley-Hamilton ideal CH(Du

D) ⊂ R ⊗A BD is still closed by Lemma 3.1.2.3, so that E(R,Du

D)

is a profinite BD-algebra. However, E(R,DuD

)/mDE(R,DuD

) ∼= (R ⊗A FD)/CH(D) is notnecessarily a finite FD-vector space, and does not necessarily carry the discrete topology.The former fact suggests that a non-topological moduli space of representations may not befinite type over SpecBD, and the latter fact implies that a non-topological moduli space ofrepresentations may not correspond to continuous representations.

Let (−)∧D

denote mD-adic completion of a BD-scheme. This is the formal completion ofa BD-scheme X at the subscheme X := X ×SpecBD

SpecFD.

Corollary 3.2.4.3. Assume ΦD. The formal Spf(A)-groupoid of representations RepD

(resp. RepD, resp. RepD) is naturally isomorphic over PsRD to the mD-adic completion ofRep

D|E (resp. RepD|E , resp. RepD|E). In particular, it is a formally finite type, Notherian

PsRD-formal scheme (resp. a formally finite type Notherian PsRD-formal algebraic stack).Additionally, the map

ψ : RepD −→ PsRD, (resp. ψ : RepD → PsRD)

pushes forward coherent sheaves to coherent sheaves and is universally closed.

Remark 3.2.4.4. Note that once we know that the Rep groupoids are representable byformal schemes/algebraic stacks, Corollary 3.2.3.4 can be used to deduce that they are adicover Spf(BD).

Proof. The isomorphism between these formal groupoids follows directly from Theorem3.2.4.1.

For the rest of the proof, we recall Corollary 1.5.4.7, which describes the properties ofψ in the non-topological case. The properties in the statement of the corollary are stableunder adic completion. The finiteness of the pushforward of coherent sheaves involves abit of work. Modulo each power mn

Dof the maximal ideal of BD, the map from RepD|E ×

SpecBD/mnD

to SpecBD/mnD

is an adequate moduli space following by a finite morphismby [Alp10, Proposition 5.2.9(3)]. Therefore the pushforward of a coherent sheaf is coherent.The pushforward of a coherent sheaf on the whole formal scheme consists of the inverse limitat each of these finite levels, and this is a finite BD module by e.g. Lemma 3.2.2.4.

Remark 3.2.4.5. The result on coherent sheaves would be more straightforward if weknew that formal GAGA holds over adequate moduli spaces. It has been recently proved inthe slightly narrower case of good moduli spaces [GZB12].

3.2.5. Consequences of Algebraization. We conclude our work on pseudorepresen-tations by applying our best results from Chapter 1 and Chapter 2 to the moduli of contin-uous representations and pseudorepresentations of a profinite algebra R over a Noetherianprofinite local ring A. In particular, we find pleasant conclusions as corollaries of

134

(1) the projective subspaces of θ-stable representations in fibers of ψ, and completelocal projective deformations of these spaces (Theorem 2.2.4.1), and

(2) the adequacy of ψ and ψ in the neighborhood of residually multiplicity free pseu-dorepresentations (Corollary 2.3.3.9).

(3) the projectivity over complete local pseudodeformation rings of moduli spaces ofrepresentations which have a certain ordering of extensions (Corollary 2.2.4.3), ver-ifying a proposal of Kisin [Kis09a, Remark 3.2.7].

Theorem 3.2.5.1. Let A be a Noetherian profinite local ring with residue field FA andlet R be a continuous A-algebra. Choose a residual pseudorepresentation D ∈ PsRd

R(FA)satisfying finiteness condition ΦD. Then

(1) All continuous representations of R over admissible A-algebras factor uniquely con-tinuously through the Cayley-Hamilton algebra E(R,Du

D), which is an algebra finite

as a module over the complete Notherian local A-algebra BD.(2) The Spf(A)-formal scheme (resp. formal algebraic stacks) of representations Rep

D

(resp. RepD, resp. RepD) are the mD-adic completion of the finite type, non-topological SpecBD-scheme (resp. algebraic stack) of representations Rep

D|E (resp.

RepD|E , resp. RepD|E), which are also continuous representations when restricted to

admissible BD-algebras. Consequently, RepD

(resp. RepD, resp. RepD) are finitetype over Spf BD.

(3) If the residual pseudorepresentation D is split over FA and is stabilizing relative toa character θ of the Grothendieck group of the abelian category of representationsof the finite dimensional FA-algebra E(R,Du

D) ⊗BD FA, there is a PsRD-projective

subscheme Reps

D(θ) of RepD parameterizing representations whose reduction modulomD is θ-stable.

(4) Assuming that D is split and multiplicity free over FA, given an ordering of thenon-isomorphic simple representations ρi, 1 ≤ i ≤ n of R over FA such that D =

det (⊕n1 ρi), there exists a PsRD-projective subscheme Rep′D ⊂ RepD of represen-

tations which are residually a certain ordering of extensions given in Definition2.2.3.2.

(5) If a d-dimensional residual representation D of R is split and multiplicity free andof characteristic greater than 2d, then ψ (resp. ψ is an adequate moduli space. Inparticular, this means that PsRD is precisely the GIT quotient of Rep

D with respectto the adjoint action.

Proof. Part (1) is Theorem 3.2.3.3. Part (2) is Corollary 3.2.4.3. For part (3), weapply Theorem 2.2.4.1 to RepD, using the fact that the base BD is complete. Part (4) is anapplication of Corollary 2.2.4.3. Part (5) is Corollary 2.3.3.9, where we use the fact that BD

is complete and therefore henselian.

In particular, this theorem can take A to be the universal deformation ring BD of aresidual pseudorepresentation of R over FA satisfying ΦD.

135

CHAPTER 4

p-adic Hodge Theory in Group-Theoretic Families

This chapter is a generalization of Mark Kisin’s Potentially Semistable DeformationRings [Kis08, §§1-2]. We also provide some additional expository content as we do this.There, the constructions start given a continuous representation of the absolute Galois groupΓ = ΓK := Gal(K/K) of K, a finite extension of Qp, on a free module over a complete Noe-therian local Zp-algebra A with finite residue field. Then loci of SpecA[1/p] such that theassociated Galois representation satisfies conditions from p-adic Hodge theory are deter-mined. Our goal is to generalize the arguments and constructions of [Kis08] to the case thatA is formally finitely generated over a complete Noetherian local ring R with finite residuefield, i.e. the quotient of a restricted power series ring R〈z1, . . . , za〉. We know from Corol-

lary 3.2.4.3 that the moduli spaces of representations RepD of Γ with a residually constant

d-dimensional pseudorepresentation D : F[[Γ]]→ F are formally finite type over the completelocal Noetherian pseudodeformation ring RD with residue field F. Because the whole modulispace of d-dimensional pseudorepresentations is semi-local with local components Spf RD inbijective correspondence with Fp-valued d-dimensional pseudorepresentations of Γ (Corollary3.1.6.13), the results of this chapter apply to the whole moduli space of representations of

Γ. This means, for example, that given the condition “semistable with Hodge-Tate weights

in [0, h],” there exists a Zariski closed subspace of Repd

Γ[1/p] parameterizing precisely theserepresentations. See Theorem 4.12.12 for the p-adic Hodge theoretic conditions for which weprove such a result.

As a concrete example of the application of this theorem, consider two crystalline rep-resentations ρ1, ρ2 of ΓK over Qp. It is well known that the subset of the vector space ofextensions of the form (

ρ1 ∗0 ρ2

)that are crystalline form a sub-vector space of ExtQp[Γ](ρ2, ρ1). Our results show that fora much wider set of conditions – e.g. potentially semi-stable of a certain Galois type, andprescribed Hodge type – the locus of extensions fulfilling this representation will be Zariskiclosed. This is a proper generalization of the results of Kisin [Kis08], since there is not

necessarily one finite field valued representation of Γ such that the entire family of extensionsreduces to it. An example of such a case is when the mod p reductions ρi are absolutelyirreducible and dimFp ExtFp[Γ](ρ2, ρ1) > 1.

We will not reference the moduli spaces and pseudorepresentations in what follows, butwill simply assume that A is a formally finite type R-algebra, where R is a complete Noe-therian local Zp-algebra with finite residue field F and maximal ideal m. We will sometimesuse α for an Artinian ring R/mn.

136

4.1. Changes in Notation from [Kis08]

For the reader familiar with the notation of [Kis08], we remark that we follow the notationthere with the following exceptions. For the most part, the changes come from generalizingthe coefficient ring of the representation, as described above.

(1) We use Γ for the Galois group denoted as GK in [Kis08] and Γ∞ for GK∞ .(2) For the portions of [Kis08] where A represents an Artinian local ring with residue

field F, and VA is a free A-module with an A-linear continuous action of Γ∞, we letA be a finitely generated over an Artinian local ring α. Here α stands in for R/mn

for some n > 0. The topology on A is the discrete (mα-adic) topology. In particular,this means that A is finite type over Z. We let VA be a projective rank d A-modulewith anA-linear action of Γ∞ with open kernel.

(3) When in [Kis08] A represents a complete Noetherian local ring with finite residuefield F, in the analogous sections of our work A will represent a formally finite typeR-algebra, i.e. a quotient of a finitely generated restricted power series ring overR. The ring R is a complete Noetherian local ring with finite residue field F. Thismakes A a topologically finite type Zp-algebra, where the topology on A is mR · A-adic. We then use VA to denote a projective rank d A-module with an action ofΓ.

(4) When [Kis08] changes notation and uses A in place of A, and then A = A[1/p], wedo the same. We also require that A be p-torsion free along with this transition,i.e. A is a flat continuous topologically of finite type Zp-algebra.

4.2. Background for Representations of Bounded E-height (§§4.3-4.5)

Let k be a finite field of characteristic p > 0 and W := W (k) its ring of p-typical Wittvectors. W is the ring of integers of a finite unramified extension K0 := W (k)[1/p] of Qp.Let K/K0 be a totally ramified extension of degree e. Fix an algebraic closure K of K, and

a completion Cp of K and let Γ := ΓK = Gal(K/K).

Our entire aim is to study the moduli of representations of Γ with p-adic Hodge theoreticproperties. We recall the definitions of some p-adic period rings.

Let OK be the ring of integers of K and OCp the ring of integers of Cp. Let R = lim←−OK/p,where each transition map is the Frobenius endomorphism of the characteristic p ring OK/p.This is a complete valuation ring which is perfect of characteristic p and whose residue fieldis k and is also canonically a k-algebra [FO, Proposition 4.6]. The fraction field FrR of R is acomplete nonarchimedean algebraically closed characteristic p field. The elements x of R arein natural bijection with sequences of elements (x(n))n≥0 of OCp such that xp(n+1) = x(n) for

all n ≥ 0. A canonical valuation on R is given by taking the valuation v on Cp normalizedso that v(p) = 1 and setting vR((x(n))n≥0) = v(x(0)). Frobenius ϕ acts on R by the pthpower map also, or, equivalently, a single shift in the limit defining R or, in terms of thepresentation x = (x(n))n≥0, ϕ(x) = (xp(n))n≥0.

Consider the ring W (R), and write an element of W (R) as (x0, x1, . . . , xn, . . . ). There isa unique continuous surjective W -algebra map

θ : W (R) −→ OCp

(x0, x1, . . . ) 7→∞∑n=0

pnxn,(n)

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lifting the projection to the first factor R → OK/p onto the 0th truncation W0(R) of thelimit of truncated Witt vectors defining W (R) (cf. [FO, Remark 5.10]). There is a Frobeniusaction on the perfect, characteristic p ring R, and therefore also a Frobenius map ϕ on W (R)which sends (x0, x1, . . . ) to (xp0, x

p1, . . . ).

We fix the notation S := W [[u]], the power series ring in the variable u. We equip Swith a Frobenius map denoted ϕ, which acts by the usual Frobenius map on W and sends uto up. We think of these as the functions bounded by 1 on the open analytic unit disk overK0, and S[1/p] as the ring of bounded functions on the open unit disk. Fix a uniformizerπ ∈ K, and elements πn

1 for n ≥ 0 such that π0 = π and πpn+1 = πn. Write E(u) ∈ W [u] forthe minimal, Eisenstein polynomial of π. We note that ϕn(E(u)) is a minimal, Eisensteinpolynomial for πn for n ≥ 0.

Write π := (πn)n≥0 ∈ R, and let [π] ∈ W (R) be its Teichmuller lift (π, 0, 0, . . . ). Becausethe R is canonically a k-algebra, we have a canonical embedding W → W (k) → W (R).We consider W (R) as a W [u]-algebra by sending u to [π]. Since θ([π]) = π, this embeddingextends to an embedding of S into W (R) (cf. the formulation of W (R) in [FO, §5.2.1]), andwe will consider W (R) and rings derived from W (R) as S-algebras via this map from nowon. From the discussion above, this map is visibly ϕ-equivariant.

We define another important element [ε] ∈ W (R). Firstly define a sequence of pnth rootsof unity

(4.2.1) ε0 = 1, ε1 6= 1, and εpn+1 = εn ∀n ≥ 0.

This sequence defines an element ε in R. Let [ε] ∈ W (R) be its Teichmulller lift. Noticethat θ([ε]− 1) = 0.

Let OE be the p-adic completion of S[1/u]. Then OE is a discrete valuation ringwith residue field k((u)) and maximal ideal generated by p. Write E for its fraction fieldFrOE = OE [1/p]. The inclusion S → W (R) extends to an inclusion OE → W (FrR), sinceπ ∈ FrR and W (FrR) is p-adically complete. Let Eur ⊂ W (FrR)[1/p] denote the maximalunramified extension of E contained in W (FrR)[1/p], and OEur its ring of integers. Since FrRis algebraically closed, the residue field OEur/pOEur is a separable closure of k((u)). If OEur isthe p-adic completion of OEur , or, equivalently, the closure of OEur in W (FrR) with respectto its p-adic topology, set Sur := OEur ∩W (R) ⊂ W (FrR). All of these rings are subrings ofW (FrR)[1/p], and are equipped with a Frobenius operator coming from W (FrR)[1/p].

For n ≥ 0 let Kn+1 := K(πn), and let K∞ = ∪n≥0Kn and Γ∞ := Gal(K/K∞). Clearly the

action of Γ∞ on W (R) fixes the subring S, since it fixes both W and πn ∀n ≥ 0. Therefore

Γ∞ has an action on Sur and Eur.The discussion above provides the needed background and definitions for §§4.3-4.5, where

“representations of E-height ≤ h” are discussed. Background and definitions for the rest ofthe chapter are given in §4.6.

4.3. Families of Etale ϕ-modules

In this section, let A denote an algebra of finite type over an Artinian local ring α withfinite residue field F of characteristic p. Let VA be a finite projective constant rank A-modulewith an A-linear action of Γ∞ with open kernel, i.e. an object of the additive exact tensorcategory

RepΓ∞(A).

1In the notation above, these would be π(n).

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Write ModΓ∞(A) for the category of finitely generated A-modules with an action of Γ∞ with

open kernel.Let OE,A denote OE ⊗Zp A, with an A-linear extension ϕ of the Frobenius on OE . We

note that this is a Noetherian ring, as OE is Noetherian and A is finitely generated over Zp.Write Φ′M(A) for the category of OE,A-modules M with an isomorphism ϕ∗(M)

∼→ M , andwrite ΦM(A) for the full subcategory of projective constant rank modules. These are knownas etale ϕ-modules over A. This is also a additive exact rigid tensor category.

Much of p-adic Hodge theory has to do with equivalences between categories of Galoisrepresentations and categories of linear algebraic data. We wish to prove an equivalence ofthis sort between the categories above.

In the case that A = α = Zp, this is due to Fontaine [Fon90, A.1.2.6], who proved thatthe following functors are quasi-inverse and therefore define an equivalence of categories:

M : ModΓ∞(Zp) −→ Φ′M(Zp)

VZp 7→ (OEur ⊗Zp V∗Zp)

Γ∞

V : Φ′M(Zp) −→ Mod(Γ∞, A)

MZp 7→ (OEur ⊗OE MZp)ϕ=1

By adding A-structure for A Artinian with residue field F (say A = α), it is immediatethat M,V extend to mutually quasi-inverse functors on the analogous abelian categorieswith A-linear structure, ModΓ∞

(A) and Φ′M(A). It is shown in [Kis09c, Lemma 1.2.7(4)]that this equivalence of categories restricts to an equivalence of the respective additive exactsubcategories of projective, finite, constant rank objects, RepΓ∞

∼= ΦM(A).Our goal in this section is to extend this theorem to the case that A is finite type over α.

We make the following definitions in order to accomplish this, also reviewing the definitionswe made at the beginning of this section.

Definition 4.3.1. Let A be a finite type α-algebra, where α is a local Artinian ring withresidue field F.

(1) Let ModΓ∞(A) be the category of finite A-modules with a A-linear action of Γ∞ with

open kernel. Let RepAΓ∞

be the full subcategory whose objects are finite, projective,and constant rank as A-modules.

(2) Let Φ′M(A) be the category of finite OE,A-modules M equipped with an A-linear

isomorphism ϕ∗(M)∼→ M . Let ΦM(A) be the full subcategory whose objects are

finite, projective, and constant rank as OE,A-modules.(3) Let M be the functor

M : ModΓ∞(A) −→ Φ′M(A)

VA 7→ (OEur ⊗Zp V∗A)Γ∞ .

(4) Let Φ′GalM (A) be the essential image of MA in Φ′M(A), and let ΦGal

M (A) be the essentialimage of M in ΦM(A).

(5) Let V be the functor

V : Φ′GalM (A) −→ ModΓ∞

(A)

MA 7→ (OEur ⊗OE MA)ϕ=1.

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Of course, it remains to be confirmed that the definition above is valid, e.g. that M(VA)is finite as a OE,A-module when VA ∈ ob ModΓ∞

(A)We note that V makes sense on all of Φ′M(A), but we only confirm after restricting it to

the full subcategory Φ′GalM (A) of Φ′M(A) that it yields an object of ModΓ∞

(A). There, weconfirm that it defines a quasi-inverse to M , making M fully faithful and exact. Thereforeit will define equivalences of categories

ModΓ∞(A)

∼−→ Φ′GalM (A),

RepΓ∞(A)

∼−→ ΦGalM (A).

In summary, this is what we want to prove.

Proposition 4.3.2 (Generalizing [Kis09c, Lemma 1.2.7]).

(1) The functor M : ModΓ∞(A)→ Φ′M(A) is exact and fully faithful, and is an equiva-

lence onto the full subcategory Φ′GalM (A) with quasi-inverse V .

(2) If A′ is a finite A-algebra, then there is a functor ΦGalM (A) → ΦGal

M (A′) induced by−⊗A A′.

(3) For W a finite A-module and VA ∈ RepΓ∞(A), there is a natural isomorphism

M(VA ⊗AW ) ∼= M(VA)⊗AW.(4) M restricts to an equivalence of categories

M : RepΓ∞

∼−→ ΦGalM (A).

In particular, this means that(a) if VA is projective as an A-module of constant rank d, then MA is a projective

OE,A-module of constant rank d, equipped with an isomorphism ϕ∗MA∼→MA.

(b) if VA is free as an A-module with rank d, then MA is a free rank d OE,A-module.

Remark 4.3.3. In the proof of this proposition, we will see that the obstruction toproving that M and V are mutually quasi-inverse on all of Φ′M(A) is that there may not be a

filtration of MA ∈ Φ′M(A) into finite α-submodules Mi such that the structure ϕ∗(MA)∼→MA

is the limit of such maps on Mi. The analogous filtration always exists in ModΓ∞(A) because

we demand that the action of Γ∞ factors through a finite quotient.

First we assemble these facts on limits. We will append (−)∞ to various categories toindicate that the A-module finiteness condition has been dropped; however, it is importantthat we do not drop the condition that the action of Γ∞ has open kernel.

Fact 4.3.4. In a category of modules, tensor products commute with direct limits, sincetensor product operations are left-adjoint functors and therefore commute with colimits.

Lemma 4.3.5. If the maps of a filtered direct limit of finite modules in Mod∞Γ∞

(α) (resp. in

Φ′∞M (α)) are all injective, then the functor (−)Γ∞ (resp. (−)ϕ=1) commutes with this direct

limit.

Lemma 4.3.6. With A as specified above, both A and OE,A are commutative JacobsonNoetherian rings.

Proof. We know that A is a Noetherian ring because it is finitely generated over anArtinian ring α with a finite residue field, and it is Jacobson because it is finitely generated

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as a Z-algebra. We observe that if pi = 0 in A, then

OE,A ∼= (W (k)/piW (k))[[u]][1/u]⊗Zp A.

Since the left factor of the tensor product is a Noetherian ring and the right factor is finitelygenerated over Zp, OE,A is Noetherian.

A commutative Noetherian ring B is Jacobson if and only if there are no primes p suchthat B/p is 1-dimensional and semi-local (see Sublemma 4.5.8). Let p be a 1-dimensionalprime of OE,A. The factor map to OE,A/p factors through the quotient ring k((u))⊗ZpA. Thisinduces a prime pc of A by contraction along the map A→ k((u))⊗Fp A/pA, and we observethat since OE,A/p is 1-dimensional, so is A/pc. Since A/pc is not semi-local and injects intoOE,A/p, neither is OE,A/p semi-local.

We also record this fact, which will be of use later.

Fact 4.3.7. Inverse limits in ModΓ∞(α) (resp. Φ′M(α)) commute with the invariant func-

tor (−)Γ∞ (resp. (−)ϕ=1), since an invariant functor is a right-adjoint functor and thereforecommutes with limits.

In order to prove the proposition above, our basic strategy will be to forget the A-linearstructure and write the objects of the categories above as direct limits of finite α-submoduleswith the respective additional structure of ϕ or a group action.

Proof (Proposition 4.3.2). Let VA ∈ ob ModΓ∞(A). Because the action of Γ∞ has

a finite index kernel, we have a canonical isomorphism as α[Γ∞]-modules of VA with lim−→iVi,

where (Vi)i∈I ∈ ob ModΓ∞(α) are the α-module-finite α[Γ∞]-submodules of VA. We note that

the functor M (resp. V ) commutes with injective direct limits in ModΓ∞(α) (resp. Φ′M(α)),

using the fact and lemma above and the fact that the tensor product⊗ZpOEur (resp.⊗OEOEur)preserve injective maps.

Therefore there are canonical isomorphisms of colimits of objects of Φ′M(α),

M(VA) = M(lim−→i

Vi) = lim−→i

M(Vi),

and the fact that M is an equivalence of categories out of ModΓ∞(α) commuting with the

necessary colimits implies that there is a canonical isomorphism respecting all structures

(4.3.8) VA ⊗Zp OEur∼

OEur ,α,Γ∞,ϕ

// M(VA)⊗OE OEur

The A-linear structure on the left hand side then provides a canonical A-linear structure onthe right hand side, commuting with the action of OEur , Γ∞, and ϕ.

Let H be the open kernel of the action of Γ∞ on VA. Since H acts trivially on VA, thecanonical isomorphism above induces a canonical isomorphism

(4.3.9) VA ⊗Zp (OEur)H∼−→M(VA)⊗OE (OEur)H .

Since Γ∞/H is finite and (OEur)Γ∞ = OE , we know that (OEur)H is finite as a OE -module.Therefore the left hand side is finite as a OE,A-module, so that the right hand side is as well.As M(VA) is a OE,A-submodule of the right hand side and OE,A is a Noetherian ring, M(VA)is finite as a OE,A-module. This confirms that the target of M can be taken to be Φ′M(A).Since V commutes with the same limits as M does, we observe that V defines a quasi-inverse

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on the essential image Φ′GalM (A) of M . This establishes (1). In particular, M is exact, since

lim−→ over a direct limit with injective maps is an exact functor on a category of modules.Part (2) is clear from the fact that (2) holds when A is replaced by α (cf. [Kis09c, Lemma

1.2.7(2)]), along with the compatibility of tensor products with direct limits of modules.For part (3), observe that this is clear for free A-modules W and then use the exactness

of M on a presentation for a general finite module W .For part (4), first observe that the exactness of M implies that M(VA) is flat over OE,A

if and only if VA is flat over A. As these modules are finite over Noetherian rings, theyare projective. Therefore, it remains only to verify that the ranks are constant, as claimed.Since both VA and M(VA) are flat, the rank function is locally constant. At a maximal idealm, we know that the ranks dimA/m VA ⊗A A/m and rkOE,A/m M(VA ⊗A A/m) are the same

since, by (2),M(VA)⊗A A/m ∼= M(VA ⊗A A/m)

and since A/m is a finite field, [Fon90, A.1.2.4(i)] tells us that the OE,A/m-rank of M(VA/m)is constant and is the same as the A/m-dimension of VA/m. Any maximal ideal I of OE,Acontains the kernel of the factor map OE,A → OE,A/m for some maximal ideal m of A.Therefore the rank of M(VA) is constant and equal to the A-rank of VA at all maximalideals. Since OE,A is Jacobson and Noetherian by Lemma 4.3.6, this means that maximalideals are dense in SpecOE,A and M(VA) has constant rank. We conclude that M(VA) is afinite, projective, constant rank OE,A-module with rank equal to rkA(VA).

We conclude by proving (4b): M(VA) is free when VA is free. The isomorphism (4.3.9)shows that both VA ⊗Zp (OEur)H and M(VA) ⊗OE (OEur)H are free (OEur,A)H-modules. ButSpec(OEur,A)H → SpecOE,A is a finite surjective etale morphism. Because vector bundlesare locally isotrivial (i.e. Hilbert theorem 90, or GLd is special in the sense of Serre [Ser58,Expose 1]), M(VA) must be free.

4.4. Functors of Lattices and Affine Grassmanians

We recall that A denotes a discrete commutative ring, finitely generated over an Artiniancommutative ring α with finite residue field F. Also, VA denotes a rank d projective A-modulewith an A-linear action of Γ with open kernel.

In the previous section, we established an equivalence between representations VA ofΓ∞ over A and certain OE,A-modules M(VA) with a Frobenius semi-linear endomorphism.Since p is nilpotent in A (say pi = 0 in A), OE,A ∼= (Z/piZ)[[u]][1/u] ⊗Zp A. ThereforeSA[1/u] ∼= OE,A, and we may consider SA-lattices withinM(VA) with a Frobenius semi-linearendomorphism inducing that on M(VA). The functor of such SB-sublattices of M(VA)⊗AB,for B a commutative A-algebra, is represented by an affine Grassmannian, as we will seebelow. An affine Grassmannian is an Ind-projective scheme, but a condition called “finite E-height,” which we will describe below, cuts out a closed subscheme that turns out to be finitetype over A. It will turn out that the condition “E-height ≤ h” corresponds to the condition“Hodge-Tate weights in [0, h]” for representations of Γ. These lattices are generalizations ofthe functor of finite flat group scheme models for VA in the case that h = 1. This was thecase studied initially in [Kis09c].

Recall that when R is a complete local ring and B is an R-algebra, RB denotes themR-adic completion of the tensor product R ⊗Zp B (so this completion will be discrete inthis section). Note also that the assumptions on A imply that A is a finitely generated

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Zp-algebra. This implies that if R (for example R = S) is Noetherian and B is a finitelygenerated A algebra, then RB is Noetherian as well. In particular, SA is Noetherian. Wewrite SB for the u-adic completion of SB, which is also Noetherian.

Definition 4.4.1. Where R → S is an injection of rings, we mean by a R-sublatticeof a projective rank d S-module M a R-submodule N of M that is projective rank d as anR-module and spans M , i.e. the natural map N ⊗R S →M is surjective.

Affine Grassmannians for inner forms of GLd are functors of sublattices of projectiveconstant rank modules. The local affine Grassmanian parameterizes these vector bundlesover the formal one-dimensional disk D which are trivialized on the punctured disk. Theglobal affine Grassmanian parameterizes these vector bundles over the affine line A1 whichare trivialized on the punctured line.

Definition 4.4.2. Let M be a projective rank d A-module. Then the affine Grassmani-ans are the following functors.

(1) The local affine Grassmanian GrlocGL(VA) for GL(VA) is the functor associating to aA-algebra B the set of pairs (PD, η) where PD is a projective rank d B[[t]]-moduleand η is an isomorphism

PD ⊗B[[t]] B((t))∼−→M ⊗A B((t)).

(2) The global affine Grassmannian GrglobGL(VA) for GL(VA) is the functor assigning to an

A-algebra B the set of pairs (PA1 , η), where PA1 is a projective rank d B[t]-moduleand η is an isomorphism

PA1 ⊗B[t] B[t][1/t]∼−→M ⊗A B[t][1/t].

We observe that there is a natural functor

(4.4.3) GrglobGL(VA) −→ GrlocGL(VA)

given by restriction from a line to the disc. Remarkably,

Theorem 4.4.4 (Beauville-Laszlo [BL95]). The functor (4.4.3) is an isomorphism.

Therefore we can call the Ind-projective scheme which represents these functors “the”affine Grassmanian. Let us overview this Ind-projective structure, and namely its canonicalample line bundle, using the local affine Grassmanian for a free module.

Recalling the definition of the local affine Grassmannian, its B-points when VA is the freemodule A⊕d amounts to the set of projective rank d B[[t]]-submodules L of B((t))⊕d whichare sublattices. For any such L, there exists some n ≥ 0 such that

(4.4.5) tn ·B[[t]]⊕d ⊆ L ⊆ t−n ·B[[t]]⊕d.

We call these lattices ti for short. Now let L be the image of L in the finite free rank 2dn B-module t−n/tn. Therefore L defines a point in some (conventional) projective Grassmannianparameterizing submodules of t−n/tn:

L ∈2dn∐k=0

PGr(k, t−n/tn)(B),

where we write PGr(k, t−n/tn) for the Grassmannian PGr(k, 2dn) of rank k projective sub-modules of a free rank 2dn B-module, identifying the lattice t−n/tn that the Grassmannian

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is constructed from. The sublattice L is a t-stable submodule, i.e. it is closed under thenatural action of t. The t-stability condition is a Zariski closed condition in this disjointunion of Grassmannians; we denote the resulting projective SpecA-scheme by

(4.4.6) PGr(t−n/tn)stable.

It turns out that any t-stable submodule L of t−n/tn lifts to a well-defined sublattice L ⊂B((t)), so that we can canonically identify PGr(t−n/tn)stable as a subfunctor of the local affineGrassmannian for A⊕d. That is, we have for each n a canonical embedding

PGr(t−n/tn)stable → GrlocGL(A⊕d)∼= GrlocGLd

.

There are also natural closed immersions

(4.4.7) PGr(t−n/tn)stable → PGr(t−n′/tn

′)stable

for all n′ ≥ n. Since, as we noted above, any L ∈ GrlocGL(A⊕d) belongs to one of these

PGr(t−n/tn)stable, we have written the local affine Grassmannian as an Ind-projective A-scheme.

There is a canonical line bundle on GrlocGL(A⊕d) which is very ample on every one of the pro-

jective subschemes PGr(t−n/tn)stable, and this is the determinant line bundle ∧dO(Grloc)(L) =

detL of the universal lattice L. Strictly speaking, the canonical line bundle is the quotientof the determinant by the determinant of the standard lattice which is, in the construc-tion above, for any A-algebra B, the lattice B[[t]]⊕d ⊂ B((t))⊕d. One can check that thisline bundle is compatible with the maps (4.4.7), and that its restriction of detL to eachof the conventional Grassmannians PGr(k, t−n/tn)stable is canonically isomorphic to the therestriction of the standard very ample line bundle on PGr(k, 2dn) to the t-stable locus.

As a result of the overview above, we can identify Ind-projective scheme GrGL(VA) andthe canonical very ample line bundle on GrGL(VA) even when VA is merely finite projectiveand not free. Of course, this could be done directly, but we will accomplish this by gluing.We may choose a Zariski cover of SpecA trivializing VA and then follow the construction ofthe Ind-projective scheme representing the affine Grassmannian for GL(VA) on this cover,along with its very ample line bundle. Since the very ample line bundle is canonical, it canbe glued together along with the Ind-scheme. Projectivity of a morphism is local on the basewhen the base is locally Noetherian and the very ample line bundle is considered to be partof the data of a projective morphism. This fact, and a discussion of notions of projectivityof morphisms, are discussed in Appendix A.

We summarize our discussion in the following

Theorem 4.4.8. Let S be a locally Noetherian scheme, and let V be a projective, coherent,constant rank OS-module. Then the affine Grassmannian GrGL(V ) is an Ind-projective S-scheme with a canonical very ample invertible sheaf arising from the determinant of theuniversal lattice.

Remark 4.4.9. For a discussion of the universal very ample determinant line bundle forthe affine Grassmannian for SLd, see [Fal03, p. 42]).

In preparation to apply the Beauville-Laszlo theorem and the affine Grassmannian to thefunctor of SA-sublattices of M(VA), we give the following proposition, which says that thefunctors of sublattices that arise in our study are sandwiched between the global and localaffine Grassmanians via (4.4.3), and therefore are all isomorphic to the affine Grassmanian.

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Some of these functors will not arise in the study below, but this proposition shows thatconsidering those functors would amount to the same thing.

Proposition 4.4.10. If VA is an object of RepΓ∞(A), projective of rank d, and MA :=

M(VA) is the corresponding OE,A-module in ΦGalM (A), then there exist equivalences between

the following functors on A-algebras.

(1) The global affine Grassmanian GrglobResW/Zp GL(VA) for ResW/Zp GL(VA)/A.

(2) The functor associating to a finitely generated A-algebra B the SB-sublattices ofMB := MA ⊗A B

(3) The functor associating to a finitely generated A-algebra B the SA⊗AB-sublattices

of (MA ⊗SA SA)⊗A B.

(4) The functor associating to a finitely generated A-algebra B the SB-sublattices of

MB := MB ⊗SB SB.(5) The local affine Grassmanian ResW/Zp GrlocGL(VA) for ResW/Zp GL(VA)/A.

Remark 4.4.11. We will see in the proof that the equivalence between the affine Grass-mannians and the functors (2), (3), (4) is not canonical. This is not a new phenomenon thatarises when A is no longer Artinian as it was in [Kis08]. There, both VA and MA were freemodules whenever they were projective of constant rank since their respective base ringsA and OE,A were semi-local, and the isomorphism between the functor of lattices of MA

and the affine Grassmannian for GLd rested on choosing non-canonical isomorphisms with astandard free module, e.g. VA ∼= A⊕d.

Proof. First let us assume that VA is free of rank d, so that MA is as well, by Proposition4.3.2(4). For simplicity we assume that W = Zp. Let B be a finitely generated A-algebra.Since p is nilpotent in A (say pi = 0) we observe that OE,B ∼= Z/piZ[[u]][1/u]⊗Zp B and

B[u][1/u] ⊂ OE,B ⊆ OE,B ⊗SB (SA ⊗A B) ⊆ OE,B ⊗SB SB ⊆ B[[u]][1/u].

Likewise, we observe that

B[u] ⊂ SB ⊆ SA ⊗A B ⊆ SB ⊆ B[[u]],

and that each member of the row above is obtained by adjoining [ 1u] to the corresponding

member of the row below.Let MA be a free A[u][1/u]-module and choose an isomorphism MA ⊗A[u][1/u] OE,A

∼→MA. The functor associating to a finitely generated A-algebra B the set of B[u]-sublattices

of MA ⊗A B is naturally equivalent to the global affine Grassmannian GrglobGLd. The local

affine Grassmannian is naturally equivalent to the functor (4). The natural transformationsbetween the functors (1) to (4) by tensoring factor the usual natural transformation fromthe global affine Grassmannian to the local affine Grassmanian. Since this transformationis known to be an equivalence by Theorem 4.4.4, all of the functors are equivalent. We notethat the choice of basis makes the equivalence between the affine Grassmannian and thefunctors of lattices non-canonical.

In the case that VA is a projective, rank d OE,A-module trivialized by a Zariski cover

Spec A → SpecA, then Proposition 4.3.2(4) implies that the same cover trivializes MA.Then one can apply descent (gluing) to the equivalences above to produce an Ind-projectivescheme parameterizing these lattices. Since finite etale morphisms induce equivalences of

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categories of locally free coherent sheaves (Hilbert Theorem 90), the isomorphism (4.3.9)shows that this Ind-projective scheme is isomorphic to GrGL(VA).

The functor of S-lattices of E-height at most h for VA is defined on the category ofA-algebras as follows. Recall that MA := M(VA).

Definition 4.4.12. For B an A-algebra, let MB = MA⊗AB; MB admits an extension ofϕ by linearity. Choose a positive integer h. A SB-lattice of E-height ≤ h is a SB-submoduleMB ⊂MB such that

(1) MB is a finite projective SB-module of rank d which generates MB as a OE,B-module, i.e. it is a sublattice.

(2) MB is stable by ϕ and the cokernel of ϕ∗(MB)→MB is killed by E(u)h.

We write L≤hVA (B) for the set of SB-lattices of E-height at most h in MA = M(VA).

Proposition 4.4.13 (Following [Kis09c, Proposition 2.1.7]). The functor L≤hVA sending acommutative B algebra to the set of SB-lattices of MB of E-height at most h is representedby a projective A-scheme L≤hVA . If A → A′ is a finite map and VA′ = VA ⊗A A′, then there

is a canonical isomorphism L≤hVA ⊗A A′ ∼→ L≤hVA . Moreover L≤hVA is equipped with a canonical

(functorial in A) very ample line bundle.

The proof is just the same as [Kis09c, Proposition 2.1.7], except that we need Proposition4.4.10 to see that the set of SB-sublattices of MB ⊂ MB is parameterized by the affineGrassmannian for ResW/Zp GL(MA) over A.

Proof. To show that L≤hVA is represented by an Ind-projective SpecA-scheme, we notethat it is naturally, by Proposition 4.4.10, a subfunctor of GrResW/Zp GL(VA). The SB-sublattice

MB ⊂MB is an object listed under (2) in Proposition 4.4.10, and therefore these sublatticesdefine points of the affine Grassmannian. The affine Grassmannian is an Ind-projectiveSpecA-scheme by Theorem 4.4.8. One can check that this subfunctor is Zariski closed, thecondition coming from the finite E-height, and therefore L≤hVA is a Ind-projective scheme. Itremains to show that this scheme is in fact finite type over A.

Choose a SA-sublattice NA ⊂ MA (with no ϕ-structure). In direct analogy with theconstruction of the Ind-projective model for GrlocGLd

out of projective subschemes (4.4.6), thecondition

unNB ⊂MB ⊂ u−nNB

(which is analogous to (4.4.5)) is a projective subscheme of GrResW/Zp GL(VA). We will complete

the proof by showing that there exists an n such that all MB of E-height ≤ h satisfy thiscondition.

In this we follow the proof of [Kis09c, Proposition 2.1.7] directly. The only modificationwe need to make is to remark that we can reduce to the case that VA and MA are free byreplacing SpecA with a Zariski cover. This reduction is possible because the affine Grass-mannian can be canonical glued together, cf. the discussion immediately before Theorem4.4.8. Let B be a finitely generated commutative A-algebra and choose MB ∈ L≤hVA (B). Letr be the least integer such that

urNB ⊂ (1⊗ ϕ)ϕ∗(NB) ⊂ u−r,

and let i be the least integer such that NB ⊂ u−iBMB. Consider a matrix which transformsa SB-basis of NB into a SB-basis of MB. From this we see that, as ϕ(u) = up, the least

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integer j such that

(4.4.14) (1⊗ ϕ)ϕ∗(NB) ⊂ u−j(1⊗ ϕ)ϕ∗(MB)

is equal to ip. Therefore we have inclusions

(4.4.15) (1⊗ ϕ)ϕ∗(NB) ⊂ u−rNB ⊂ u−i−rMB = E(u)−hu−i−r(E(u)hMB).

Recall that e denotes the degree of E and let s be the least integer such that ps = 0 in A.Now because E(u) = ue + pf(u) for some f(u) ∈ W [u] of degree e− 1,

E(u)−1 =1

ue + pf(u)=

u−e

1 + u−epf(u)= u−e(1− pu−ef(u) + · · ·+ (−1)s−1u−e(s−1)f(u)s−1).

Therefore E(u)−h · N ⊂ u−ehsN for any S-lattice N . We also know that E(u)h ·MB ⊂(1⊗ ϕ)ϕ∗(MB) because MB has E-height ≤ h, by definition. Therefore (4.4.15) extends toan inclusion

(1⊗ ϕ)ϕ∗(NB)(4.4.15)⊂ E(u)−hu−i−r(E(u)hMB) ⊂ u−i−r−ehs(1⊗ ϕ)ϕ∗(MB).

Combining this inclusion with the fact that ip is the least integer satisfying (4.4.14) meansthat

ip ≤ ehs+ i+ r, i.e. i ≤ ehs+ r

p− 1.

On the other hand, if i is the least integer such that MB ⊂ u−i(1⊗ ϕ)ϕ∗(NB), then

(1⊗ ϕ)ϕ∗(MB) ⊂MB ⊂ u−iNB ⊂ u−i−r(1⊗ ϕ)ϕ∗(NB),

by definition of r. Then since ip is the least integer satisfying (4.4.14), we have

ip ≤ i+ r, i.e. i ≤ r

p− 1.

To summarize, we first showed that if we set n = b ehs+rp−1c, then unNB ⊂ MB. Then

we showed that n is large enough so that MB ⊂ u−nNB, and in fact the lesser numberbr/(p − 1)c would work in place of n. Therefore n has the desired property that for any

lattice MB of E-height ≤ h in MB, unNB ⊂ MB ⊂ u−iNB. This shows that L≤hVA is finitetype and projective, as desired.

To get the equivalence L≤hVA ⊗A A′ ∼→ L≤hVA , firstly we recall Proposition 4.3.2(2-3), which

implies that M(V ⊗A A′) ∼= M(VA) ⊗A A′. Then the fact that the affine Grassmannian is

compatible with base change, i.e. GrGL(VA) ×SpecA SpecA′∼→ GrGL(VA⊗AA′), completes the

proof.Finally, the canonical very ample line bundle on L≤hVA arises by restriction from the canon-

ical very ample line bundle on the affine Grassmannian, cf. Theorem 4.4.8 and the discussionpreceding it.

Write ΘA for the projective map ΘA : L≤hVA → SpecA. Write M for the universal sheaf of

Θ∗A(SA)-modules on L≤hVA and M for its u-adic completion.Now we prove a generalization of [Kis08, Lemma 1.4.1], showing that the global sections

of the universal SA-lattice in MA, with its Frobenius semi-linear structure, can recover VA ina similar fashion to the correspondence between VA and MA = M(VA) in Proposition 4.3.2,but without simply recovering MA from its S-sublattice and using Proposition 4.3.2.

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Lemma 4.4.16 (Following [Kis08, Lemma 1.4.1]). Set A := ΘA∗(OL≤hVA). There is a

canonical A-linear Γ∞-equivariant isomorphism

(4.4.17) VA := VA ⊗A A∼−→ HomSA,ϕ

(ΘA∗(M),SurA

).

In the case that A is Artinian, this was proved in [Kis08, Lemma 1.4.1]. An importantinput to this argument is the result of Fontaine [Fon90, B.1.8.4], showing that if N is a finiteS-module with a Frobenius semi-linear isomorphism of bounded E-height, then the naturalZp[Γ∞]-linear map

(4.4.18) HomS,ϕ(N,Sur) −→ HomS,ϕ(N,OEur).

induced by the inclusion Sur → OEur is an isomorphism. When N has A-linear structurethen taking A-linear maps induces a canonical A[Γ∞]-linear isomorphism

(4.4.19) HomSA,ϕ(N,SurA )

∼−→ HomSA,ϕ(N,OEur,A).

When A is Artinian with finite residue field and N is finite as a SA module, then N is finiteas a S-module. Our contribution is to generalize the argument of [Kis08, Lemma 1.4.1] whenthis is not the case.

Proof. Let M∗A denote the OE,A-dual of MA := M(VA), equipped with the induced

structure of an object of ΦGalM (A). Using the canonical isomorphism

HomOE,A(M∗A,OEur,A) ∼= MA ⊗OE,A OEur,A

and applying (−)ϕ=1 to the canonical isomorphism (4.3.8), we have a canonical isomorphism.

(4.4.20) VA∼−→ (M∗

A⊗OE,A OEur,A)ϕ=1 ∼−→ HomOE,A,ϕ(MA,OEur,A)

We want to show that the rightmost factor of (4.4.20) and the rightmost factor of (4.4.17)

are canonically Γ∞-equivariantly isomorphic.Note that ΘA∗(M) is a finite ΘA∗Θ

∗A(SA) = S⊗A A-module, and the ϕ-semi-linear OL≤hVA

-

linear endomorphism of M descends to ΘA∗(M) with E-height ≤ h: for upon applying ΘA∗to a linear endomorphism ϕ∗(M)→M we have a SA-linear map

ΘA∗(ϕ∗(M) −→ ΘA∗(M).

By pre-composing this map with the natural map ϕ∗(ΘA∗(M)) → ΘA∗(ϕ∗(M)) (which is

an isomorphism because ϕ is finite and flat as a morphism SpecS→ SpecS), we have therequired structure

ϕ∗(ΘA∗(M))∼−→ ΘA∗(ϕ

∗(M)) −→ ΘA∗(M).

By the projection formula, we have

ΘA∗(M)⊗S OE∼−→ ΘA∗(Θ

∗A(MA))

∼−→MA ⊗A A = MA

The SA-linear map ΘA∗(M) → MA, x 7→ x ⊗ 1 is injective because it is the global sectionsof the canonical injection M →MA ⊗A OL≤hVA

.

Choose now some Vi ⊂ VA, an α[Γ∞]-submodule, finite as an α-module (i.e. an object of

ModΓ∞(α)), such that the natural map Vi ⊗α A → VA is an surjection. Clearly such a Vi

exists, since VA is finitely generated as an A-module and the action of Γ∞ factors througha finite quotient. Let Mi = M(Vi) ⊂ M(VA) = MA be the corresponding OE,α-submodule,an object of Φ

′GalM (α); by Proposition 4.3.2, this is naturally a submodule and the canonical

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Φ′GalM (A)-morphism Mi ⊗α A→MA is surjective. Let N be the intersection

N := ΘA∗(M) ∩Mi ⊂MA,

which we observe is a Sα-submodule of MA. We have the natural surjection N ⊗α A ΘA∗(M).

Now the result of Fontaine (4.4.18) discussed above makes for the isomorphism (4.4.19),which we repeat here:

HomSα,ϕ(N,Surα )

∼−→ HomSα,ϕ(N,OEur,α).

Thinking of A as an α-module for a moment, applying ⊗αA to this isomorphism induces anisomorphism

HomSα,ϕ(N,SurA

)∼−→ HomSα,ϕ(N,OEur,A).

Then tensor-Hom adjunction results in an isomorphism

HomSA,ϕ(N⊗α A,Sur

A)∼−→ HomSA,ϕ

(N⊗α A,OEur,A).

Finally, because the map SurA→ OE,A inducing this isomorphism may be checked to be

an injection, an element of the left hand side factors through the quotient ΘA∗(M) if andonly if its image on the right hand side factors through ΘA∗(M). As all of the maps inthis construction were canonical, this completes the construction of the desired canonicalA[Γ∞]-linear isomorphism

HomSA,ϕ(ΘA∗(M),Sur

A)∼−→ HomOE,A,ϕ(MA,OEur,A).

Now, extending the results of [Kis08, §1.5] where A is taken to be a complete local ringwith finite residue field, we extend the above situation to mR-adic limits. Let A be a finitetype (in the sense of formal schemes) R-algebra, compatibly with the representation VA andits induced determinant. This means that A is complete and separated with respect to themRA-adic topology. As in [Kis08], for any Zp-algebra R we denote by RA for the mR-adiccompletion of R⊗Zp A.

The functor M generalizes to this setting naturally from the above, since

(4.4.21) MA = (OEur⊗ZpV∗A)Γ∞ ∼→ lim←−(OEur ⊗Zp V

∗A ⊗A A/(mRA)i)Γ∞

by Fact 4.3.7. This isomorphism follows from the fact that inverse limits commute withinvariant functors, and the ideal (p⊗A)+(OEur⊗mRA) (with which the left side is completed)is equal to OEur⊗ZpmRA (with which the right side is completed). This means that MA is aprojective OE,A-module of rank d as expected.

For B an A-algebra such that miR ·B = 0 for some i ≥ 1, set L≤hVA (B) = L≤hVA/(mRA)i

(B).

Corollary 4.4.22. The functor L≤hVA on A-algebras B such that miR · B = 0 for some

i ≥ 1 is represented by a projective A-scheme L≤hVA .

Proof. By Proposition 4.4.13, this functor is represented by a projective formal schemewith a very ample line bundle compatible with its limit structure. By applying formal GAGA(perhaps locally and gluing) we conclude that L≤hVA is the mR-adic completion of a projectiveA-scheme.

It will be useful later to know that SA is Noetherian. This is the main technical use ofthe condition that A is finite type (in the sense of formal schemes) over R.

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Lemma 4.4.23. The formal scheme Spf(SA) is Noetherian.

Proof. Firstly, we claim that SR is Noetherian. This is the case because of two facts:the standard complete tensor product −⊗Zp− is the tensor product in the category of com-

plete Noetherian local rings with finite residue fields. Therefore S⊗ZpR is Noetherian (seee.g. [Gro64, 0IV, Lemme 19.7.1.2]). It is also isomorphic as a ring (though not necessarily asa topological ring) to SR, the mR-adic completion of S⊗Zp R, because the residue fields ofboth S and R are finite, and therefore SR is Noetherian.

Now note that SA, defined to be the mR-adic completion of S ⊗Zp A, is isomorphic to

SR⊗RA, where this completed tensor product is taken in the category of adic R-algebras,i.e. this is the categorical dual of the fiber product of formal Spf(R) schemes. BecauseSpf(A)/ Spf(R) is finite type and Spf(SR) is Noetherian, [Gro60, Proposition 10.13.5(ii)]implies that Spf(SA) is Noetherian.

4.5. Universality of M in Characteristic 0

We now study the image of the map ΘA : L≤hA → SpecA in characteristic 0, i.e. afterinverting p, following [Kis08, §1.6]. We will study these properties through their points infinite local W (F)[1/p]-algebras B, and therefore will need to study S-modules or OE -moduleswith coefficients in such rings B. Therefore very little new is needed in addition to [Kis08]to accomplish this. The main new content is Lemma 4.5.6, which is needed in order to drawconclusions about ΘA[1/p] by its behavior on finite W (F)[1/p]-algebras alone.

Remark 4.5.1. We are venturing outside the realm of linearly topologized rings in con-sidering W (F)[1/p]-algebras. For example, there is no filtered set of ideals giving a basis ofthe p-adic topology on Qp around 0 since all ideals are trivial!

However, even big rings like A[1/p] are still Noetherian. For A is the quotient ofZp[[t1, . . . , ta]]〈z1, . . . , zb〉 for some a, b ≥ 0 by the Cohen structure theorem (see e.g. [MR10,Theorem 3.2.4]), and this ring is Noetherian since is is the (p, t1, . . . , ta)-adic completion ofZ[t1, . . . , ta, z1, . . . , zb]. Then A[1/p] is Noetherian by the Hilbert basis theorem.

The preparatory Lemmas 4.5.2, 4.5.3, and 4.5.4 require no modification from [Kis08].

Lemma 4.5.2 ([Kis08, Lemma 1.6.1]). Let B be a finite Qp-algebra, and MB a finiteSB∼= S⊗Zp B-module, which is flat over S[1/p] and equipped with a map ϕ∗(MB)→MB

whose cokernel is killed by E(u)h. Suppose that E ⊗S[1/p] MB is finite free over E ⊗Qp B.Then MB is a finite projective SB-module.

The statement proof is identical to that of [Kis08], so we have omitted the proof. Thesame is true of the next two results.

Lemma 4.5.3 ([Kis08, Lemma 1.6.2]). Let B be a finite Qp-algebra, and J ⊂ K0[[u]]B =K0[[u]] ⊗Qp B be an ideal such that ϕ(J)K0[[u]]B = J , where ϕ acts B-linearly. Then J isinduced by an ideal of B.

Corollary 4.5.4 ([Kis08, Corollary 1.6.3]). Let A be a finite flat Zp-algebra, and VA a

finite free A-module equipped with a continuous action of Γ∞. Set MA := (OEur⊗ZpV∗A)Γ∞.

Suppose that VA, considered as a Zp[Γ∞]-module, is of E-height ≤ h, and let MA ⊂ MA bethe unique S-lattice of E-height ≤ h.

Then MA is a SA-submodule of MA, and MA ⊗Zp Qp is finite projective over SA[1/p].

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In the following proposition, we we work in characteristic zero, translating the uniquenessof S-lattices of E-height ≤ h into a statement about ΘA (part (1)) and showing that thescheme theoretic image of ΘA has the property we expect (part (2)).

Proposition 4.5.5 (Following [Kis08, Proposition 1.6.4]). Let A and VA be as specifiedabove. Then

(1) The map ΘA : L≤hVA → SpecA is a closed immersion after inverting p.

(2) If A≤h is the quotient of A corresponding to the scheme-theoretic image of ΘA, thenfor any finite W (F)[1/p]-algebra B, a continuous A→ B factors through A≤h if andonly if VB = VA ⊗A B is of E-height ≤ h.

Proof. Omitted. All of the elements of the proof of Proposition 4.5.5 are entirely local,but depend on the fact that A[1/p] is Jacobson with residue fields of closed points finite overQp, and that the image of A lies in the ring of integers of the residue fields. This propertyof S[1/p] when S is a complete Noetherian local Zp-algebra, and this is what is used in[Kis08]. Lemma 4.5.6 will show that A[1/p] has this property even though A is no longerlocal. Otherwise the proof requires no modification from that of [Kis08], so we omit it.

The following lemma is the main new content needed to generalize [Kis08, Proposition1.6.4] to Proposition 4.5.5. It is well-known, and quoted and deduced from [Gro66, §§10.4-10.5] in what follows.

Lemma 4.5.6. Let A be a finite type (in the sense of formal schemes) R-algebra, whereR is a complete Noetherian local Zp-algebra. Then

(1) A[1/p] is Jacobson and Noetherian,(2) all residue fields of maximal ideals are finite extension of Qp, and(3) the image of A in any such residue field is contained in its ring of integers.

We develop some notation that will be used in the proof of Lemma 4.5.6 and record afew basic facts about these notions in Sublemma 4.5.8.

Definition 4.5.7. Let R be a commutative ring.

(1) If R is a domain, we call it a Goldman domain if its fraction field is finitely generatedover itself.

(2) A prime ideal p ⊂ R is called a Goldman prime ideal provided that R/p is a Goldmandomain, i.e. provided that the residue field κ(p) of p is finitely generated over R/p.

(3) We call R a Hilbert ring provided that every Goldman prime ideal is maximal.

The following facts will be useful in proving Lemma 4.5.6.

Sublemma 4.5.8. Let R be a commutative ring.

(1) R is Jacobson if and only if R is Hilbert.(2) A Noetherian Goldman domain that is not a field must be of height 1 and have

finitely many prime ideals.(3) The fraction field of a Goldman domain R can be generated by one element over R.

Proof. Parts (1), (2), and (3) are proved in [Gro66, Proposition 10.4.5], [Gro66, Propo-sition 10.5.1], and [Gro66, Proposition 10.4.4] respectively.

Proof. (Lemma 4.5.6) Invoking the Cohen structure theorem, we can write R as acontinuous quotient of Zp[[t1, . . . , ta]]. Then, as A is finite type over R, we can find a surjection

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from Zp[[t1, . . . , ta]]〈z1, . . . , zb〉 to A. We will replace A with Zp[[t1, . . . , ta]]〈z1, . . . , zb〉 and showthat it has the desired property.

First we show that all residue fields of all maximal ideals are finite extensions of Qp. LetB = [[t1, . . . , ta]] ⊂ A. First we address the case that A = Zp[[t1, . . . , ta]]. Choose a maximalideal m ⊂ A[1/p], and let F = A[1/p]/m be the residue field with ω representing the quotientmap. Since A is complete with respect to the I-adic topology where I = (p, t1, . . . , ta), themaximal ideals of A are contractions of the maximal ideals of A/I ∼= Fp[z1, . . . , zb]. Inparticular, these maximal ideals have finite, characteristic p residue fields. But since Fhas characteristic zero, the image of ω(A) ⊂ F must not be a field and the canonical mapZp → ω(A) is injective. This means that F is generated by 1/p over ω(A), so that ω(A) is aGoldman domain by definition. Therefore, by Sublemma 4.5.8(2), ω(A) is a Krull dimensionone Noetherian domain containing Zp and having finitely many primes. This domain is alsocomplete with respect to ω(I). As ω(I) is not (0) and ω(A) has dimension 1, its radicalr(ω(I)) must be maximal. Therefore ω(A) is a r(ω(I))-adically complete local Noetheriandomain of dimension 1. By Noether’s normalization lemma for complete Noetherian mixed-characteristic local rings [MR10, Theorem 3.2.4], ω(A) is finite as a module over a subringisomorphic to Zp[[s1, . . . , sd − 1]] where d = dimω(A). Therefore ω(A) is finite over Zp andis the ring of integers of F , a finite extension of Qp. This proves parts (2) and (3) of thelemma.

Now we show that A[1/p] is Jacobson and Noetherian. Indeed, A[1/p] is Noetherian(cf. Remark 4.5.1. Now, because p is in the Jacobson radical of A and A is Noetherian, wemay directly apply [Gro66, Corollaire 10.5.8] to say that A[1/p] is Jacobson.

Now we replicate [Kis08, Corollary 1.7]. Much of the work in [Kis08] goes through in thesame way, except the construction of MA≤h .

Proposition 4.5.9 (Following [Kis08, Corollary 1.7]). There exists a finite SA≤h-moduleMA≤h such that

(1) MA≤h is equipped with a map ϕ∗(MA≤h)→MA≤h whose cokernel is killed by E(u)h.(2) MA≤h ⊗Zp Qp is a projective SA≤h [1/p]-module.(3) For any finite W (F)[1/p]-algebra B, any map h : A≤h → B and any C ∈ IntB

through which h factors, there is a canonical, ϕ-compatible isomorphism of S⊗ZpB-modules

MA≤h ⊗A≤h B∼→MC ⊗C B.

(4) There is a canonical isomorphism

VA≤h ⊗Zp Qp∼→ HomS

A≤h [1/p],ϕ(MA≤h ⊗Zp Qp,SurA≤h [1/p]).

Proof. Recall that SA is the mR-adic completion of S ⊗Zp A and is Noetherian by

Lemma 4.4.23. Let L≤hVA be the mR-adic completion of L≤hVA . Then

ΘSA : L≤hVA ×Spf A Spf SA → Spf SA

is a projective morphism of Spf(A)-formal schemes over a Noetherian formal scheme. The

mR-adic completion M of M may be regarded as a formal coherent (further, locally free)

sheaf on L≤hVA×Spf ASpf SA. Applying formal GAGA to ΘSA (which requires that SA be Noe-

therian), M is the completion of a coherent (further, locally free) sheaf M on the projectiveSA-scheme

ΘSA : SpecL≤hVA ×SpecA SpecSA → SpecSA.

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The scheme theoretic image of ΘSA is SA≤h . We set

MA≤h := ΘSA∗(M).

With this work done, the proofs of part (1), (2), (3) and (4) may be repeated from [Kis08,Corollary 1.7].

Part (1) results from the fact that M has the desired map ϕ∗(MA≤h)→MA≤h , and that,as ϕ is a flat map on S, ϕ∗ commutes with direct images. Part (2) follows from the factthat ΘA is the identity operator after p is inverted, and M is a locally free coherent sheaf onL≤hA ×SpecA SpecSA. Part (3) builds on Proposition 4.5.5(2) and its proof. The statementof Proposition 4.5.5(2) tells us that VC = VA ⊗A C is of E-height ≤ h, which means thatMC = M(VC) contains a unique S-lattice of E-height ≤ h, MC . Now consider the imageof MA≤h ⊗A≤h C in MC : this is a torsion free, ϕ-stable SC-submodule of MC such that thecokernel of ϕ∗(M′

C) → M′C is killed by E(u)h. Following the proof of Proposition 4.5.5(2),

this implies that OE ⊗S M′C ∩M′

C [1/p] is a S-lattice of E-height ≤ h in MC , and thereforeis equal to MC . This shows that M′

C ⊗C C[1/p] ∼= MC ⊗C C[1/p], from which the statementof part (3) follows.

For part (4), we use Lemma 4.4.16: Let A := ΘSA∗(OL≤hVA). We observe that there is a

canonical isomorphismVA

∼→ HomSA,ϕ(ΘSA∗(M),Sur

A)

by combining Lemma 4.4.16, which implies this statement for A replaced by A/miR, and

the theorem on formal functions. Applying formal GAGA, inverting p, and noting thatProposition 4.5.5 implies that A≤h[1/p]

∼→ A[1/p], we get a canonical isomorphism

VA≤h ⊗Zp Qp = VA ⊗Zp Qp∼→ HomSA,ϕ

(MA≤h ⊗Zp Qp,SurA

[1/p]).

Since the map A≤h → A has p-torsion kernel and cokernel, the same is true of SurA≤h → Sur

Aand SA≤h → SA, completing the proof.

4.6. Background for Families of Filtered (ϕ,N)-modules (§§4.7-4.12)

Following [Kis08, §2], we change notation, now denoting with A the adic R-algebra Afrom above. We now assume that A is p-torsion free, i.e. flat over Zp, and write A forA[1/p], which (Lemma 4.5.6) is Jacobson with residue fields of maximal ideals finite overQp.

For R a Zp-algebra we write RA := RA [1/p], where we recall that RA is the mRA-adiccompletion of R⊗Zp A

. We extend ϕ to an A-linear endomorphism of SA. We will use the

canonical isomorphism SA/uSA∼→ WA

∼= W [1/p]⊗Qp A.Let O := lim←−n(W [[u, un/p]][1/p]), which we may think of as the ring of rigid analytic

functions on the open disk of radius 1, including S[1/p] the dense subring of boundedfunctions. The Frobenius endomorphism ϕ has a unique continuous extension from S[1/p]to each ring W [[u, un/p]][1/p], and therefore to O as well.

Let c0 = E(0) be the constant coefficient of the Eisenstein polynomial for π, and set

λ :=∞∏n=0

ϕn(E(u)/c0) ∈ O

Denote by S0 the completion of K0[u] at the ideal (E(u)).

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In order to study families over A of ϕ-modules over O, we need to define the correctnotion of the ring of coefficients. In fact, two candidate definitions end up being the same:

OA := lim←−n

(W [[u, un/p]]A) ∼= lim←−n

(WA [[u, un/p]][1/p].

While it is clear that these rings are isomorphic when A is local, we prove the isomorphismhere in the general case.

Lemma 4.6.1. The natural inclusions

W [[u, un/p]]A → WA [[u, un/p]][1/p]

induce an isomorphism

OA := lim←−n

(W [[u, un/p]]A)∼−→ lim←−

n

(WA [[u, un/p]][1/p]).

Proof. Write Bn := W [[u, un/p]]A and Cn := WA [[u, un/p]][1/p], with the canonical map

Bn → Cn that we get from considering an element of Bn as a power series in u. Since themaps making up these limits are injective, it will suffice to show for f ∈ C2n that its imagein Cn under the inclusions making up the limit lies in the image of Bn in Cn. With f ∈ C2n

chosen, write it as

f =∑m≥0

fmum

pbm/2nc

where fm ∈ p−NA ⊆ A for some fixed N ≥ 0. This expression also denotes the naturalimage of f in Cn under inclusion. We rewrite it as

f =∑m≥0

fmpbm/nc−bm/2nc um

pbm/nc.

We want to show that f lies in the image of Bn in Cn. This is the case because the coefficient

fmpbm/nc−bm/2nc of un/pbm/nc lies in m

i(n)A where limn→+∞ i(n) = +∞; this is the case because

p ∈ mA .

We observe that SA → OA, and we extend ϕ to an A-linear endomorphism of OA as it

was for O above. Write S0,A for the completion of K0[u]⊗Qp A at the ideal (E(u)).Now, following [Kis08, §2.3], we consider period rings over the base A. Firstly, we recall

the period rings themselves (the basic case A = Zp). Let Acris be the p-adic completion of thedivided power envelope of W (R) (see §4.2) with respect to ker(θ), and let B+

cris := Acris[1/p].The action of ϕ on W (R) extends to an action on Acris [FO, §6.1.2]. The map S[1/p] → B+

cris

extends uniquely to an continuous inclusion O → B+cris, because S[1/p] is dense in O, and

the eth power of the image [π] of u in W (R) is in the divided power ideal (ker θ, p) for Acris

[FO, Proposition 6.5] (for more detail on the radius of rigid analytic functions appearing inB+

cris, see Lemma 4.6.6).Define B+

dR to be the ker(θ)-adic completion of W (R)[1/p], where θ is extended to a mapθ : W (R)[1/p] Cp, and let BdR be its fraction field. This is a discrete valuation ringwith residue field Cp and maximal ideal ker θ and BdR is its valuation field, but the topologyof B+

dR as a (complete) discrete valuation ring does not agree with its topology induced bythe constructions we have made so far, and we use the latter topology. General theory ofcharacteristic zero complete local rings implies that θ has a section, but there is no choice of

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section that is Γ-equivariant, nor is there a section which is continuous. The rings Acris, B+cris

are canonically subrings of B+dR.

Recall from §4.2 the definition of [π], the image of u in W (R), and [ε] of (4.2.1). The“logarithms” of these elements are important elements of B+

dR, which we now define.Write `u ∈ B+

dR for

`u = log [π] :=∞∑i=1

(−1)i−1

i

([π]− ππ

)i.

This series converges in B+dR because θ([π] − π) = π − π = 0, and B+

dR is by definitionthe ker θ-adic completion of W (R)[1/p]. Because θ([ε] − 1) = 0 and B+

dR is the ker(θ)-adiccompletion of W (R)[1/p], the series

t = log[ε] :=∞∑i=1

(−1)i+1 ([ε]− 1)i

i

converges to an element in B+dR, which we call t. In fact, because the denominators in this

series are sufficiently bounded in terms of the powers of ([ε]− 1), one can show that t ∈ Acris

and tp−1 ∈ pAcris [FO, Proposition 6.6].Using `u and t, we can define several more period rings: Bcris := B+

cris[1/t] ⊂ BdR,B+

st := B+cris[`u] ⊂ B+

dR, and Bst := Bcris[`u], which we can think of as a polynomial ringbecause `u is transcendental over the fraction field of Bcris [FO, Proposition 6.11]. As both`u and t are “logarithms,” it is not hard to see that ϕ(`u) = p`u and ϕ(t) = pt, so we extendϕ to these rings according to those rules.

Equip B+st with an endomorphism N , by formal differentiation of the variable `u with

coefficients in B+cris, i.e. so that N(B+

cris) = 0. Extend ϕ to B+st as well, with ϕ(`u) = p`u.

We note that ϕ and N define endomorphisms of the polynomial subring K0[`u] ⊂ B+st , with

N again acting by formal differentiation with respect to the variable `u. We observe thatpϕN = Nϕ on Bst. Neither ϕ nor N extend continuously to an action of B+

dR (cf. [FO,Remark 5.18(3)]); only the filtrations that we will now describe come from B+

dR.There is an exhaustive, decreasing filtration on each of Acris, B

+cris, written

FiliAcris,FiliB+cris

where Fil0Acris = Acris (resp. Fil0B+cris = B+

cris) induced by their inclusion in the filtered ringB+

dR. The filtration on B+dR is given by

FiliB+dR := (ker θ)i, i ≥ 0.

In fact, t ∈ Fil1B+dR and t 6∈ Fil2B+

dR [FO, Proposition 5.19], so also t ∈ Fil1Acris, and t isa generator for the maximal ideal of B+

dR. We note that the associated graded ring of B+dR

may be represented as Cp[t], and, when we naturally extend the filtration to BdR∼= B+

dR[1/t],grBdR

∼= Cp[t, 1/t]. We also note that

(4.6.2) FiliAcris · Filj Acris ⊆ Fili+j Acris,

and similarly for B+cris.

Now we discuss the action of Γ on these period rings. The action arises from the actionof Γ on OK/p, and extends from there to continuous actions on R, W (R), and all of the

subrings of BdR defined above. That B+st is stable under Γ follows from Lemma 4.6.4 below,

where we calculate the action of Γ on `u, finding that for σ ∈ Γ that σ(`u) differs from `u by

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a product of an element of Zp and t. It will also be useful to know the action of Γ on t: wesee that

σ(εn) = εχn(σ)n ,

where χn : Γ → Z/pnZ is the reduction modulo pn of the p-adic cyclotomic character

Zp(1) = χ : Γ → Zp. We find that σ(ε) = εχ(σ), and a calculation with the “logarithm”defining t = log [ε] tells us that

(4.6.3) σ(t) = χ(σ) · t.Now we calculate the Galois action on `u. For σ ∈ Γ, define β(σ) ∈ B+

cris as

β(σ) := σ(`u)− `u.

Lemma 4.6.4. The map β is a 1-cocycle with respect to the cyclotomic character, belong-ing to the cohomology class associated to π by Kummer theory. When β(σ) 6= 0, it generatesthe maximal ideal of B+

dR.

Proof. For σ ∈ Γ and n ≥ 1, define ηn(σ) ∈ Z/pnZ by the relation

σ(πn) = εηn(σ)n · πn.

As πpn+1 = πn and εpn+1 = εn, we see that ηn+1(σ) ≡ ηn(σ) (mod pn) and we have a well

defined map η : Γ → Zp. We observe that η is a cocycle for the cyclotomic character

χ : Γ→ Z×p , because

εηn(τσ)n =

τσ(πn)

πn=τ(ε

ηn(σ)n · πn)

πn=εχn(τ)ηn(σ)n · εηn(τ)

n

πn= εχn(τ)ηn(σ)+ηn(τ)

n .

A change in the choice of roots of unity (εn) amounts to a change in η by a coboundary forχ; the same is true for a new choice of pnth roots (πn) of π. This η is the definition of the

“Kummer cocycle” in H1(Γ,Zp(1)) induced by π under the map

K×/(K×)pn ∼−→ H1(Γ,Zp/pnZp(1)), n ≥ 1

coming from the long exact sequence in Galois cohomology, which is an isomorphism byHilbert’s Theorem 90.

We now see that σ(π) = π ·εη(σ). Therefore σ([π]) = [π]·[ε]η(σ), and one can quickly verifythat even though `u = log [π], t = log [ε] are not standard logs, we still have the expectedidentity

σ(log [π]) = log [π] + η(σ) · log [π].

Now β is given in terms of η:

β(σ) = σ(`u)− `u = log [π] + η(σ) · log [π]− log [π] = η(σ) · t.It is also clear that β(σ) = 0 if and only if σ ∈ Γ∞. Therefore we see that because t ∈ Acris, we

have for any σ ∈ Γ that β(σ) ∈ Acris, and when β(σ) 6= 0, then β(σ) generates the maximalideal ker θ of B+

dR because t is a generator and ker θ/(ker θ)2 ∼= Cp as a Zp-module.

As this maximal ideal generates the filtration on B+dR, if β(σ) 6= 0 then

(4.6.5) β(σ) ∈ Fil1B+cris, β(σ) 6∈ Fil2B+

cris.

Having completed our summary of the period rings we will need, we now explain theconstruction of period rings with coefficients in A.

156

Define B+cris,A := Acris,A [1/p], where Acris,A is, as usual, the mRA

-adic completion of

Acris ⊗Zp A. For any A-algebra B, we write B+

cris,B for B+cris,A ⊗A B. Set B+

st,A := B+cris,A[`u]

and B+st,B := B+

st,A ⊗A B. The map ϕ extends to each of these rings B-linearly, with N

again acting as formal differentiation with respect to `u here. In particular, N(B+cris,B) =

0. Analogous notation is used for the elements of the filtration on these rings: denoteby FiliAcris,A the mRA

-adic completion of FiliAcris ⊗Zp A, and for any A-algebra B let

FiliB+cris,B := FiliAcris,A ⊗A B. Basic properties over Zp, mainly (faithful) flatness of both

period rings and graded pieces of their filtrations, are extended to these period rings andfiltrations with coefficients in Lemmas 4.8.1 and 4.8.2.

It will be important to know in the construction of (4.9.4) that there is a canonicalinclusion OA → B+

cris,A extending the map O → B+cris discussed above, and also a map

SurA → B+

cris,A. By Lemma 4.6.1, it will suffice to show that for large enough n,

W [[u, un/p]]A → Acris,A .

In order to construct the map, it will suffice to draw, for sufficiently large n, maps

W [[u, un/p]]⊗Zp A/(mRA

)j → Acris ⊗Zp A/(mRA

)j

for each j ≥ 1. We will get such maps if we show, for large enough n, the existence of maps

W [[u, un/p]] → Acris.

Then Lemma 4.8.1 implies that this map will remain injective after tensoring with A andcompleting with respect to the mRA

-adic topology. This same construction gives us acanonical map Sur

A → B+cris,A.

In fact we will show much more than this, which will be useful later (e.g. the proof ofLemma 4.9.9).

Lemma 4.6.6. For r ∈ (0, 1) let Or denote the coordinate ring of the open rigid analyticdisk over K0 with radius r. Then for any r > (e(p − 1))−1, O → B+

cris factors through Or.In particular, W [[u, un/p]] → Acris for n > e(p− 1).

Proof. Recall that u is sent to [π] in Acris, and Acris is the p-adic completion of thedivided power envelope of W (R) with respect to ker θ. In fact, the kernel of θ, which isdefined as the composition

Acrisθ−→ OCp OCp/p,

is also a divided power ideal of Acris [FO, Proposition 6.5]. Recall that θ([π]) = π and thatπ is the uniformizer of an extension K of Qp with ramification degree e, so that [π]e is inthe divided power ideal. Since denominators m! may accompany the image of powers of uas small as uem, and the p-adic valuation of m! satisfies vp(m!) ∼ m/(p − 1) as m → +∞,we see that Acris is a W [[u, ua/pb]]-algebra whenever a/b > e(p− 1).

4.7. Families of (ϕ,N)-modules over the Open Unit Disk

Following the conclusion of §4.5, we assume that we are given a finite projective SA-module MA of constant rank d with a ϕ-semilinear, A-linear endomorphism ϕ : MA →MA

such that the induced SA-linear ϕ∗(MA)→MA has cokernel killed by E(u)h. We write

MA := MA ⊗SA OA, DA :=MA/uMA,

each of which have a natural induced action of ϕ.

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Lemma 4.7.1 ([Kis08], Lemma 2.2). There is a unique, ϕ-compatible, WA-linear mapξ : DA →MA, whose reduction modulo u is the identity on DA.

The induced map DA ⊗WAOA → MA has cokernel killed by λh, and the image of the

map DA ⊗WAS0,A →MA ⊗OA S0,A is equal to that of

ϕ∗(MA)⊗OA S0,A →MA ⊗OA S0,A.

The proofs of [Kis08, Lemma 2.2 and Lemma 2.2.1] go through verbatim. None ofit depends on A being local. They are a generalization of [Kis06, Lemma 1.2.6], whereA = Zp.

Proof. Let s0 : DA →MA be any WA-linear section of the projectionMA → DA. Ourcandidate for the map ξ is the sum

s = s0 +∞∑i=0

(ϕi+1 s0 ϕ−i−1 − ϕi s0 ϕ−i).

We claim that s converges to a well-defined WA-linear map from DA toMA. Accepting this,we see immediately that s is equivalent to the identity modulo u and that ϕ s = s ϕ, asdesired.

Let DA ⊂ DA be a finitely generated WA-submodule which spans DA. Similarly, wechoose a finitely generated SA-submodule M

A ⊂ MA which spans MA. We may chooseM

A so thatϕ s0 ϕ−1 − s0 : DA −→ uMA

takes DA into uMA. Acting on this map by ϕ, we find that

ϕi+1 s0 ϕ−i−1 − ϕi s0 ϕ−i : DA → upi

MA

as well. Choose j ≥ 0 with the property that ϕ induces a map MA → p−jM

A and ϕ−1

induces a map DA → p−jDA. Then for each i ≥ 0, we have

ϕi+1 s0 ϕ−i−1 − ϕi s0 ϕ−i : DA → p−2ijupi

MA.

Because the exponential outraces the polynomial, for arbitrarily large n ≥ 0 this seriesconverges to a well defined map

DA −→MA ⊗SA WAW [[u, up

n

/p]]A,

and therefore to a map ξ : DA →MA.To see the uniqueness of ξ, we argue following [Kis06, Lemma 1.2.6]. Given another

map ξ′ satisfying the stipulations of the statement of the lemma, we consider the imageX = (ξ − ξ′)(DA) of (ξ − ξ′). The image lies in uMA because both maps are sections of the

projection onto DA. Then because ϕ is an automorphism of DA, X ⊂ ϕj(uMA) ⊆ upjM

for all j ≥ 0. Therefore X ∼= 0 as desired.Note that ξ reduces to the identity modulo u and that its determinant, being an element

of OA, may be safely said to belong to W [[u/p]]A. As a result, Lemma 4.7.4 tells us that forsufficiently large n ≥ 0, ξ induces an isomorphism

(4.7.2) DA ⊗WAW [[u/pn]]A

∼−→MA ⊗OA W [[u/pn]]A.

Denote by ξs the map

DA ⊗WAW [[up

s

/pn]]A −→MA ⊗OA W [[ups

/pn]]A.

induced by ξ.

158

Consider the commutative diagram

ϕ∗(DA ⊗WAOA)

ϕ∗ξ //

∼

ϕ∗MA

1⊗ϕ

DA ⊗WAOA

ξ //MA

Let r be the least integer such that e < pr/n. Applying ⊗OAW [[u, ups/pn]]A to the diagram

above for s = 0, . . . , r− 1 yields a diagram where the right vertical arrow is an isomorphism,because π ∈ K× lies outside the radius of convergence of some of the elements of W [[u, up

s/pn]]

when e ≥ ps/n, and the cokernel of this map is supported at π. Using the fact that thevertical arrows send u to up and ξ is ϕ-equivariant, we see that if we tensor the top row byW [[up

s/pn]]A to get ξs, we may tensor the lower row by W [[up

s+1/pn]]A to get ξs+1:

(4.7.3) DA ⊗WAW [[up

s/pn]]A

∼

ξs //MA ⊗OA W [[ups/pn]]A

1⊗ϕ

DA ⊗WAW [[up

s+1/pn]]A

ξs+1 //MA ⊗OA W [[ups+1/pn]]A

We now have a visible argument by induction with base case (4.7.2) that ξs is an isomor-phism for s = 0, 1, . . . , r − 1.

Now consider the case for s = r − 1, where the radius of convergence will contain π butnot π1/p. Now the top row of (4.7.3) will be an isomorphism, but the right side verticalarrow will may have non-trivial cokernel killed by E(u)h. We also see the final claim of thelemma, which is that the image of the right vertical arrow coincides with the image of thelower horizontal arrow formally locally around π.

Repeating the argument as above shows, for any s ≥ 0, that the cokernel of ξs is killedby∏s

i=r ϕs−r((E(u)/c0)h). Therefore, recalling the definition of OA, we see that λh kills the

cokernel of ξ ⊗WAOA.

Lemma 4.7.4 ([Kis08, Lemma 2.2.1]). Let I ⊂ W [[u]] be a finitely generated ideal suchthat IW [[u/p]]A/uW [[u/p]]A is the unit ideal. Then for n sufficiently large, IW [[u/pn]]A is theunit ideal.

Proof. Suppose first that I is principal, equal to (f) for f ∈ W [[u/p]]A. The assumptionsimply that the image of f in W [[u/p]]A/uW [[u/p]]A

∼= WA is a unit. Because there is a naturalinjection WA → W [[u/p]]A, f may be multiplied by a unit in W [[u/p]]A so that its image inW [[u/p]]/uW [[u/p]]A = WA is 1. In particular, f ∈ 1 + uW [[u/p]]A. There exists some j ≥ 0such that f ∈ 1 +p−jW [[u/p]]A since W [[u/p]]A = W [[u]]A [1/p] by definition. Therefore f hasan inverse in W [[u/pj+1]]A . This is the desired n of the statement of the lemma.

In general, write I = (f1, . . . , fr) for f1, . . . , fr ∈ W [[u/p]]A. Write fi for the image offi in W [[u/p]]A/uW [[u/p]]A. Then 1 =

∑ri+1 gifi for some gi ∈ W [[u/p]]A/u. Choose lifts

gi ∈ W [[u/p]]A of the gi. Then by the first part,∑r

i=1 gifi is a unit in W [[u/pn]] for somesufficiently large n.

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4.8. Period Rings in Families

We will now record some lemmata to ensure that the large rings B+cris, Acris, and so forth

behave well in families. This will allow us to show later that, for example, (ϕ,N)-modulesin families behave as one would expect (cf. Theorem 4.10.9).

Lemma 4.8.1 ([Kis08], Lemma 2.3.1). For any A-module M , denote by M its mR-adiccompletion. If M is a flat A-module, then

(1) For any finite A-module N , the natural map

N ⊗A M → N ⊗A Mis an isomorphism.

(2) M is flat over A. If M is faithfully flat over A, then so is M .

(3) The functor M 7→ M preserves short exact sequences of flat A-modules.

For completeness, we elaborate on the proof in [Kis08].

Proof. First we claim that the functor on finite A-modules N 7→ N ⊗A M is exact.Say that we have an exact sequence of finite A-modules

0→ N1 → N2 → N3 → 0.

The Artin-Rees Lemma implies that the filtrations (mRA)n · Ni have bounded difference,

i.e. there exists k such that for all n ≥ k,

(mRA)n ·N1 ⊆ ((mRA

)n ·N2) ∩N1 ⊆ (mRA)n−kN1,

and the filtrations on N2 and N3 are easily seen to be equal. This implies that the na-tive mRA

-adic topologies on N1 and N3 are equivalent to the topologies induced by themRA

-adic topology on N2. Therefore completion with respect to these topologies maintainsexactness (AM Cor. 10.3). We claim that since M is flat, tensoring this exact sequenceand these adic filtrations by M preserves the bounded difference of the filtrations. Indeed,because M is flat, for any ideal I of A and A-module N , (I ·N)⊗A M ∼= I · (N ⊗A M).Therefore the composition of the −⊗AM -functor with the (mRA

)-adic completion functoris exact as desired.

To see (1), let N be a finite A module and observe that (1) is obvious when N is free.For N a general finite A-module, we take a presentation by free modules Fα → Fβ and find

Fα ⊗A M //

o

Fβ ⊗A M //

o

N ⊗A M

// 0

o

0 // Fα ⊗A M // Fβ ⊗A M // N ⊗A M // 0

The five-lemma shows that N ⊗A M∼→ N ⊗A M as desired.

The first part of (2) follows from the fact that the injectivity of the map I ⊗A M →Mis preserved under mRA

-adic completion. For the second part, if M is faithfully flat overA if and only if Mm 6= 0 for all maximal ideals of the (formal scheme corresponding to) A.This property is clearly preserved under completion.

(3) follows from the same considerations on filtrations discussed above. If N1 ⊆ N2 areflat A-modules, then the native mRA

-adic filtration on N1 and the filtration induced fromthe native filtration on N2 are equal.

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Lemma 4.8.2 (Following [Kis08, Lemma 2.3.2]).

(1) For i ≥ 0, the ideal FiliAcris,A of Acris,A is a faithfully flat A-module.(2) For i ≥ 0, FiliAcris,A is a faithfully flat A-module, which is isomorphic to the

mR-adic completion of (FiliAcris/Fili+1 Acris)⊗Zp A.

(3) For any A-algebra B, i ≥ 1, and σ ∈ Γ, B+cris,B/(β(σ)B + FiliB+

cris,B) is a flat

B-module. If β(σ) 6∈ FiliB+cris, then β(σ) 6∈ FiliB+

cris,B.(4) Let B be a finite continuous A-algebra with ideal of definition I. Then the I-adic

completion of Acris ⊗Zp B is canonically isomorphic to Acris,A ⊗A B.

(5) The map

Acris,A →∏

Acris,A/q

is injective, where q runs over ideals of A such that A/q is a finite flat Zp-algebra.(6) If 0 6= f ∈ Acris, then f is not a zero divisor in Acris,A.

Proof. For part (1), since FiliAcris is a faithfully flat Zp-module, FiliAcris ⊗Zp A is a

faithfully flat A-module. Then Lemma 4.8.1(2) implies that FiliAcris,A is a faithfully flatA-module.

To demonstrate part (2), take the exact sequence of faithfully flat Zp-modules

0→ Fili+1Acris → FiliAcris → FiliAcris/Fili+1Acris → 0,

apply the exact functor ⊗ZpAcris, and the mRA-adic completion functor. The latter functor

is exact by Lemma 4.8.1(3) and preserves the condition “fully faithful” by 4.8.1(3). Thedesired result is then visible in the resulting exact sequence of faithfully flat A-modules.

To prove part (3), first consider the case β(σ) = 0. Using the logic of part (2) and applyingit by induction on i to Acris/FiliAcris, thinking of it as an extension of Acris/Fili−1Acris byFili−1Acris/FiliAcris, we find that Acris,A/FiliAcris,A is a flat A-module. We may thenapply ⊗AB for any A-algebra B to the exact sequence

0→ FiliAcris,A → Acris,A → Acris,A/FiliAcris,A → 0

to get the result.Now allow β(σ) 6= 0. Let j be the largest integer such that β(σ)/pj ∈ Acris/FiliAcris ⊂

(Acris/FiliAcris)[1/p] ∼= B+cris/FiliB+

cris. Then Acris/(β(σ)/pj · Zp + FiliAcris) will be Zp-flat(cf. the argument after (4.12.4)). We then apply the reasoning of the proof of part (2) andthe first case of (3) found above to the exact sequence of flat Zp-modules

0→ Zp·β(σ)/pj−→ Acris/FiliAcris → Acris/(β(σ)/pj · Zp + FiliAcris)→ 0

in order to conclude that B+cris,B/(β(σ) ·B + FiliB+

cris,B) is a flat B-module.It remains to prove the final statement in part (3). Because tensor products and direct

limits commute, we may assume that B is a finitely generated A-algebra. Then there existsa map B → B′, where B′ is a finite W (F)[1/p]-algebra. We show the contrapositive: Ifβ(σ) ∈ FiliB+

cris,B, then β(σ) ∈ FiliB+cris,B′ as well, which implies that β(σ) ∈ FiliB+

cris

(because B′ is trivially a flat Qp-algebra).For part (4), the fact that f : A → B is finite and continuous implies that f is adic,

i.e. that f(mRA) · B, like the ideal I of B,, is an ideal of definition for B. So we simply

take I to be f(mRA) · B. Then because B is a finite A-algebra, the mRA-adic topology

on Acris,A ⊗A B is equivalent to its I-adic topology by the Artin-Rees lemma. Now we

161

apply Lemma 4.8.1(1) to conclude that the natural map

Acris,A ⊗A B → lim←−n

(Acris ⊗A B)⊗B B/In

is an isomorphism as desired.Now we prove (5). Let M be the set of maximal ideals of Spf(A) as a Spf(Zp)-formal

scheme, corresponding to maximal ideals of A/mRA. For m ∈ M , let Acris,Am denote the

completion of Acris ⊗Zp A at 1⊗m. First, we will reduce to the case that A is a complete

local Noetherian ring by showing that the natural map

(4.8.3) Acris,A −→∏m∈M

Acris,Am

is injective. Therefore we may assume that the connected components of SpecA/mRA are

positive-dimensional.The map

(4.8.4) A →∏m∈M

Am

is injective and flat, since A is Noetherian and A/mRA is Jacobson since it is finitely

generated over Z. Since Acris is a flat Zp-module, Lemma 4.8.1(2) implies that that the map

Acris,A −→ Acris,A ⊗A∏m∈M

Am

is injective. To complete the proof that the map (4.8.3) is injective, we will show that forany 0 6= f ∈ Acris,A , there exists a finite subset Mf ⊂ M such that the image of f in∏

m∈MfAcris,Am is not 0.

Having chosen a nonzero f ∈ Acris,A , there exists some n ≥ 1 such that the image of f ,call it f , in Acris ⊗Zp A

is not 0. Write out f in tensor form as

f =∑j

gj ⊗ hj

where gj ∈ Acris/pnAcris (resp. hj ∈ A/(mRA

)n) are Zp-linearly independent in Acris/pnAcris

(resp. hj ∈ A/(mRA)n), i.e. such that this tensor product cannot be reduced. Let H ⊂

A/(mRA)n be the Zp-linear span of hj. Because (4.8.4) is injective and we are assuming

that the connected components of SpecA/mRA are positive dimensional, for each nonzero

h ∈ H, there exist infinitely many m ∈ M such that there exists a power k = k(m, h) of

m such that the image of h in Acris ⊗Zp A/((mRA)n + mk(m,h)) is not zero. Since there are

only finitely many hj, this means that there are finitely many powers of maximal idealsmkm such that the images of hj in Acris ⊗Zp A

/((mRA)n + ∩mkm) remain Zp-linearly

independent. We set Mf to be this finite set of maximal ideals. Since Acris is p-adicallycomplete, the corresponding statement for the gj is trivial: we need only make sure thatpn−1 remains nonzero, which we can do by increasing each km to at least n. Now we haveshown that the reduced tensor form of f in Acris⊗Zp A

/((mRA)n +∩mkm) remains reduced,so that it cannot vanish. Therefore the image of f in

∏Mf

Acris,Am has nonzero image in∏Mf

Acris,Am/(mAcris,Am)km , and is nonzero as desired.

Now we prove (5) in the case that A is a complete local Noetherian Zp-algebra withmaximal ideal m and finite residue field, simply adding more detail to the proof in [Kis08],

162

Lemma 2.3.2. First we deduce that for any n ≥ 1, there exists q ⊂ A such that A/q isa finite flat Zp-algebra and q ⊆ mn. For since A[1/p] is Jacobson with maximal ideals aand residue fields Ka being finite extensions of Qp with rings of integers Oa, there exists aninjective map

A −→∏a

Oa

such that the kernel consists of nilpotents (recall the standing assumption that A is a flatas a Zp-algebra). Let N be the nilradical of A. Choose a finite set S of maximal ideals ofA[1/p] such that the induced map

(A/N) −→∏a∈S

Oa

is injective after tensoring with (A/N)/((mn+N)/N). Such an S exists because of the finitelength of (A/N)/((mn + N)/N). Now choose representatives in A of the finite cardinalityset N/(N ∩mn). Then there exists a finite set of powers aka of maximal ideals a of A[1/p]such that the image of these representatives in

∏aA[1/p]/aka does not vanish. Now we

observe thatA −→

∏a∈S

A/(A ∩ aka)

is injective after tensoring with A/mn, showing that the ideal

q := A ∩⋂a∈S

aka

of A satisfies the desired condition: the quotient A/q is a finite flat Zp-algebra, and q ⊆ mn.Now choose 0 6= f ∈ Acris,A . Then there exists n ≥ 1 such that the image of f in

Acris ⊗Zp A/mn is not zero. Using the ideal q constructed above, we see that the image of

f in Acris ⊗Zp A/mn is nonzero as desired.

To show (6), we use (5) to reduce to the case that A is a finite flat Zp-algebra. ThenAcris,A

∼= Acris⊗ZpA because both rings have the p-adic topology. Now (6) follows from the

flatness of A over Zp, considering that the injective map from Acris to itself by multiplicationby 0 6= f ∈ Acris remains injective after tensoring with A.

Lemma 4.8.5 (Generalization of [Kis08, Lemma 2.3.3]). Let M be an A-module andx ∈ Acris,A ⊗A M . The set of A-submodules N ⊂ M such that x ∈ Acris,A ⊗A N has asmallest element N(x).

Here we cannot repeat the proof of [Kis08], for that proof only covers the case that A

is local.

Proof. Assume that mnR ·M = 0 for some n ≥ 1. Therefore there is a natural isomor-

phism of A-modulesAcris,R ⊗RM

∼→ Acris,A ⊗A M.

Applying [Kis08, Lemma 2.3.3] to the right hand side, there exists a smallest R-submoduleP of M such that x ∈ Acris,R ⊗R P . We claim that the image N of the natural map

P ⊗R A →M

is the smallest A-submodule of M with the required property. Clearly it contains x. If therewere a A submodule N ′ with the property, then N ′ ⊃ P since N ′ is also a R-module with

163

the property. But then N ′ must contain N , which is the A-span of P . This shows that Nis the smallest A-submodule of M with the property.

Now we allow M to be any finite A-module. Now the proof may be completed accordingto [Kis08, Lemma 2.3.3]. The same is true for the generalization to an arbitrary A moduleM .

4.9. Period Maps

Following [Kis08, §2.4], recall our situation: A is an adic and finite type R-algebra,

where the structure map is compatible with the action of Γ on a projective, rank d A-module VA . Suppose that (A)≤h = A. Write MA for the finite projective SA-moduleof given by Proposition 4.5.9, with a map ϕ∗(MA) → MA with cokernel killed by E(u)h.Also set MA := MA ⊗Zp Qp. Write VA := VA ⊗A A, so that Proposition 4.5.9(4) provides

a canonical, Γ∞-equivariant isomorphism

(4.9.1) ι : VA∼→ HomSA,ϕ(MA,S

urA ).

We will follow [Kis08, §2.4] in deducing a period map from the data above. This gives us

a Γ∞-equivariant comparison with coefficient ring A between the Galois representation VAand the periods of MA. We will then discuss additional data needed in order to extend thisto a Γ-equivariant map, although this additional data ends up simply being a restriction(Proposition 4.9.11). In what follows, B is an arbitrary A-algebra.

We deduce from the map ι a SA-linear, ϕ-equivariant map

(4.9.2) MA → HomA(VA,SurA ); m 7→ (v 7→ 〈m, ι(v)〉).

Tensoring this map by ⊗SAOA and using the map ξ : DA → MA from Lemma 4.7.1, wehave a ϕ-equivariant map

(4.9.3) DAξ→MA → HomA(VA,S

urA )⊗SA OA → HomA(VA, B

+cris,A).

Tensoring the composition of these maps by ⊗AB for B our chosen A-algebra, there is aB+

cris,B-linear map

(4.9.4) DB ⊗WBB+

cris,B → HomA(VA, B+cris,A)⊗A B ∼= HomB(VB, B

+cris,B).

We see that the right hand side has an action of Γ, and the left hand side has an action ofΓ∞ through the action on B+

cris,B. This map is Γ∞-equivariant because Γ∞ acts equivariantly

on the inclusions S → O → B+cris and that ι above is Γ∞-equivariant. In order to extend

the action of Γ∞ on the left hand side of (4.9.4) to an action of Γ, we suppose that there isa WB-linear map

N : DB → DB

which satisfies the identity pϕN = Nϕ. Then the action of Γ on DB ⊗WBB+

cris,B is

(4.9.5) σ(d⊗ b) =

(∞∑i=0

N i(d)⊗ β(σ)i

i!

)σ(b) = exp(N ⊗ β(σ)) · d⊗ σ(b)

164

for σ ∈ Γ. Then we observe that such an action of Γ commutes with ϕ, using the fact thatϕ(β(σ)) = pβ(σ). Here is the calculation:

σϕ(d⊗ b) = exp(N ⊗ β(σ))ϕ(d)⊗ σ(ϕ(b))

=

(∞∑i=0

N i(ϕ(d))⊗ β(σ)i

i!

)ϕ(σ(b))

=

(∞∑i=0

piϕ(N i(d))⊗ β(σ)i

i!

)ϕ(σ(b))

=

(∞∑i=0

ϕ(N i(d))⊗ ϕ(β(σ)i)

i!

)ϕ(σ(b))

= ϕσ(d⊗ b).Now we set up a theory for semistable representations, recalling that we adjoin `u to

B+cris,B to get B+

st,B = B+cris,B ⊗K0 K0[`u], naturally extending the actions of ϕ and N to the

tensor product, with N acting as dd`u

on K0[`u]. Consider the composite of the isomorphisms

(4.9.6) DB ⊗K0 K0[`u]∼→ (DB ⊗K0 K0[`u])

N=0 ⊗K0 K0[`u](`u 7→0)⊗1−→ DB ⊗K0 K0[`u]

where the first map is the inverse to the natural isomorphism

(DB ⊗K0 K0[`u])N=0 ⊗K0 K0[`u]

∼→ DB ⊗K0 K0[`u]

induced by polynomial multiplication in K0[`u]. Tensoring (4.9.6) by B+cris,B over WB and

tensoring (4.9.4) by K0[`u] over K0, we obtain the composite map

(4.9.7) DB ⊗WBB+

st,B

(4.9.6)−→ DB ⊗WBB+

st,B

(4.9.4)−→ HomB(VB, B)⊗B B+st,B.

We claim that (4.9.4) is Γ-equivariant if and only if (4.9.7) is equivariant when Γ isregarded as acting trivially on DB. A key observation is that the an inverse to the bijection

(D⊗K0 K[`u])N=0 `u 7→0−→ D is given by d 7→ exp(−N ⊗ `u) · d. We give the calculations in the

form of this lemma.

Lemma 4.9.8. The map

DB ⊗WBB+

st,B −→ DB ⊗WBB+

st,B

given by tensoring (4.9.6) by ⊗WBB+

cris,B is Γ-equivariant, when Γ is given the trivial actionon DB on the left side and the action of (4.9.5) on the right.

Proof. We write f, g for the isomorphisms

g : D −→ (D ⊗K0 K0[`u])N=0

d 7→ exp(−N ⊗ `u) · d.f : (D ⊗K0 K[`u])

N=0 −→ D

`u 7→ 0

To see that g is an inverse to f note that any element of (D ⊗K0 K[`u])N=0 has the form∑

i≥0

di`iu, where idi +N(di−1) = 0 ∀ i ≥ 1,

165

so that it is determined by the coefficient d0, i.e. f is an injection. Now observe that

g(d0) =∑j≥0

(−1)jN i(d0)`juj!

,

which certainly has the correct form with constant coefficient, so that f and g are inverse.Now we wish to show that g ⊗K0 B

+st,B is Γ-equivariant with Γ-actions defined in the

statement of the lemma. Since this map is clearly equivariant on elements of B+st,B, it will

suffice to show for d0 ∈ D (letting B = Qp for simplicity) that

g σ(d0) = σ g(d0),

where σ acts according to (4.9.5)on the left side and with the trivial action on the right side.First we calculate the left hand side. We will use the fact that that N(β(σ)) = 0 for all

σ ∈ Γ, since β(σ) ∈ B+cris. Or one can use the chain rule: since N commutes with the action

of Γ,N((σ(`u)− `u)i) = i(σ(`u)− `u)i−1(σ(N(`u))−N(`u)) = 0

since N(`u) = 1. Now here is the calculation:

g σ(d0) := g

(∑i≥0

N i(d0)⊗ β(σ)i

i!

)

=∑i,j≥0

(−1)jN j(N i(d0))⊗ β(σ)i`jui!j!

,

which is sent by f to d0, since f kills `u and β(σ) = σ(`u)− `u. We observe that this result,namely f g σ(d0) = d0, is exactly the same as f σ g(d0), where σ is given the trivialaction in this second expression. Since f and g are mutually inverse, we have completed theproof.

Lemma 4.9.9 (Following [Kis08, Lemma 2.4.6]). For each A-algebra B, the maps (4.9.4)and (4.9.7) are injective, and their cokernels are flat B-modules.

As usual our proof follows the proof in [Kis08], adding some additional exposition andmaking the points of generalization clear.

Proof. First we note that it suffices to prove the assertions only for (4.9.4), and forB = A. For if there is an exact sequence of A-modules 0→ N ′ → N → N ′′ → 0 and N ′′ isflat, then this sequence remains exact after applying − ⊗A B for any A-algebra B. It alsosuffices to prove the statement for (4.9.4) alone, since (4.9.6) is an isomorphism.

Recall Lemma 4.8.2(5), which states that

Acris,A →∏q

Acris,A/q

is injective, where q varies over ideals ofA such that A/q is a finite flat Zp-algebra. Applyingthis lemma, we may confine ourselves to the case that A is a finite flat Zp-algebra.

We are remanded to the case that A is local, so we can repeat the proof of injectivityfrom [Kis08, Lemma 2.4.6]. Recall that (4.9.4) as defined in (4.9.3): it is the composition ofthe map ξ constructed in Lemma 4.7.1 with the map (4.9.2), tensored up through the mapsSA → OA and OA → B+

cris,A.

166

Consider the commutative diagram

SurA ⊗SA MA

// HomA(VA,SurA )

OEur,A [1/p]⊗SA MA

// HomA(VA,OurE,A [1/p])

where the top map is obtained from (4.9.2). From (4.4.21), we know that the bottom rowis an isomorphism. The left arrow is injective, because the finiteness of A over Zp gives usthat this arrow is obtained by applying −⊗SMA to the canonical inclusion Sur → OEur , andMA is flat as a S-module. This means that the top map must be injective. Furthermore,it is an injective map of finite free Sur

A -modules of equal rank and remains injective aftertensoring by −⊗Sur B+

cris. Therefore the map

MA ⊗OA B+cris,A

∼→MA ⊗SA B+cris,A → HomA(VA, B

+cris,A)

obtained from (4.9.2) by tensoring up to B+cris is injective.

It remains to address the map ξ : DA → MA, which induces the first factor in (4.9.4).From the first part of Lemma 4.7.1, we know that the determinant of

(4.9.10) DA ⊗WAB+

cris,A

ξ⊗1−→MA ⊗OA B+cris,A

is a divisor of λs for some positive integer s. However, the image in B+cris of each of the factors

ϕn(E(u)/c0, n ≥ 1 of λ are units in B+cris because the p-adic radius of convergence of these

functions with respect to u is (e(p − 1))−1 (Lemma 4.6.6). Since the zeros of ϕn(E(u)/c0)have p-adic valuation (epn)−1, they lie outside this radius of convergence for n ≥ 1. Thereforethe determinant of (4.9.10) is supported at the ideal (E([π])) is a divisor of E([π])s. Now asE([π]) is not a zero divisor in Acris, neither is it a zero divisor in B+

cris,A. This completes ourproof of the injectivity statement.

It remains to show that the cokernel of (4.9.4), appearing in the exact sequence

0→ DA ⊗WAB+

cris,A

(4.9.4)−→ HomA(VA, B+cris,A) −→ coker→ 0,

is flat as an A-module. Because an A-module M is flat if and only if TorA1 (A/I,M) = 0for all ideals I of A and TorR1 (A/I,HomA(VA, B

+cris,A)) = 0 for all finitely generated ideals

I ⊂ A because HomA(VA, B+cris,A) is flat, the cokernel will be flat if and only if (4.9.4) remains

injective after tensoring with A/I for any finitely generated ideal I of A. This is what wewill now prove.

If we had started our proof with A/I in the place of A, we would still have the injectivitystatement for A/I, just as we proved it for A above. This almost completes our proof, forwe want to show that

DA ⊗WAB+

cris,A ⊗A A/I(4.9.4)⊗AA/I−→ HomA(VA, B

+cris,A)⊗A A/I

is injective, and we know that

DA/I ⊗WA/IB+

cris,A/I

(4.9.4)−→ HomA/I(VA/I , B+cris,A/I)

is injective.One can check that there is a natural isomorphism DA⊗AA/I

∼→ DA/I , with an implicitchoice of I ⊂ A such that A/I[1/p] ∼= A/I needed to draw the map. It remains to show

167

that that natural map B+cris,A ⊗A A/I → B+

cris,A/I is an isomorphism. This is precisely what

Lemma 4.8.2(4) tells us, completing the proof.

Proposition 4.9.11 (Following [Kis08, Proposition 2.4.7]). The functor which to an A-algebra B assigns the collection of WB-linear maps N : DB → DB which satisfy pϕN = Nϕand such that (4.9.4) is compatible with the action of Γ is representable by a quotient Ast ofA.

This proof requires very little modification from that of [Kis08].

Proof. Since we will require a basis for VA, we replace A with the coordinate ring of atrivialization of VA if necessary, and note that the constructions to produce Ast can be gluedtogether.

We consider the functor sending an A-algebra B to the set of WB-linear maps

N : DB → DB

satisfying pϕN = Nϕ. This is a closed condition on the representable functor B →EndB(DA ⊗A B). Write AN for the representing A-algebra, which is of finite presentationover A.

Write ψB for the map of (4.9.4). Let B = AN with the universal map N : DAN → DAN ,

and let Γ act on DAN ⊗WANB+

cris,ANaccording to (4.9.5). For any d ∈ DAN and σ ∈ Γ, let

δσ(d) measure the failure of ψAN to commute with the action of Γ as follows:

δσ(d) := ψAN (σ(d))− σ(ψAN (d)).

Then δσ(d) ∈ Q := HomAN (VAN , B+cris,AN

). Fix a B+cris,AN

-basis for Q, and let x1, . . . , xrdenote the coordinates of δσ(d) with respect to this basis. Applying Lemma 4.8.5 withM = AN and x = xi for i = 1, . . . , r, we obtain A submodules N(xi) ⊂ AN . Let Iσ,d ⊂ AN

be the ideal generated by the N(xi), so that Iσ,d is the smallest ideal I ⊂ AN such thatδσ(d) ∈ IQ. We take

Ast := AN/∑σ,d

Iσ,d,

where d runs over all elements of DAN and σ over Γ. Clearly if B = Ast with the induced Nfrom AN , then ψAst is Galois -equivariant. We must show that this property holds functoriallyon A-algebras.

If B is an A-algebra, then a map AN → B factors through Ast if and only if the kernel Kcontains Iσ,d for each σ ∈ Γ, d ∈ DAN . Since Q is faithfully flat over AN by Lemma 4.8.1(2),it is the same to ask that for all σ, d that Iσ,dQ ⊂ KQ, or equivalently that δσ(d) ∈ KQ forall σ, d. As noted above, this last condition amounts to saying that ψB is compatible with theaction of Γ. Hence Ast represents the functor as claimed in the statement of the proposition.Now we must show that SpecAst is a closed subscheme of SpecA, which we will accomplishby showing that it is a proper monomorphism. This remaining work is no different thanwhat was done in [Kis08, Proposition 2.4.7], so we include it only for convenience.

To show that SpecAst → SpecA is a monomorphism, we show, given two maps N,N ′ :DB → DB satisfying the conditions of the proposition, that they are equal. Since Lemma4.9.9 provides a canonical injection into a Galois module, the Galois action on DB⊗WB

B+cris,B

induced by N in (4.9.5) is identical to that of N ′. Thus for σ ∈ Γ, d ∈ DB, we have an equality

d = exp(N ⊗ β(σ)) · exp(−N ′ ⊗ β(σ)) = exp((N −N ′)⊗ β(σ))d

168

of elements of DB ⊗WBB+

cris,B. Recall from (4.6.5) that β(σ) ∈ Fil1B+cris,B. Recalling also

the multiplicativity of the filtration (4.6.2), we see that

(N −N ′)(d)β(σ) ∼= 0 modulo Fil2B+cris,B.

Then by (4.6.5) and the second part of Lemma 4.8.2, β(σ) 6∈ Fil2B+cris,B whenever σ 6∈ Γ∞,

so N = N ′ as desired.Now we will check the valuative criterion of properness. Suppose that the A-algebra

B is a discrete valuation ring with uniformizer πB. Let N : DB[1/πB ] → DB[1/πB ] be an

endomorphism satisfying the conditions of the proposition. Let σ ∈ Γ be such that β(σ) 6= 0.If d ∈ DB, then

(4.9.12) exp(N ⊗ β(σ)) · d ∈ DB ⊗WBB+

cris,B[1/πB] ∩ HomB(VB, B+cris,B),

using (4.9.4) to consider both DB ⊗WBB+

cris,B[1/πB] and HomB(VB, B+cris,B) as a subset of

HomB[1/πB ](VB[1/πB ], B+cris,B[1/πB ]). Since the cokernel of (4.9.4) is flat over B by Lemma

4.9.9, it has no πB-torsion so that the intersection in (4.9.12) is the isomorphic image ofDB ⊗WB

B+cris,B.

Therefore, modulo the ideal Fil2B+cris,B ⊂ B+

cris,B,

d− exp(N ⊗ β(σ))d ∼= −N(d)⊗ β(σ) ∈ DB ⊗WBB+

cris,B/Fil2B+cris,B.

The first part of Lemma 4.8.2(3) may be used to show that N(d) ∈ DB as follows: inthis diagram, (where we use Fil2 as an abbreviation for Fil2B+

cris,B or Fil2B+cris,B[1/πB ] as

appropriate, write F = B[1/πB], and we assume σ 6∈ Γ∞)

0

0

DB

·β(σ)

// DF

·β(σ)

DB ⊗WBB+

cris,B/Fil2

// DF ⊗WFB+

cris,F/Fil2

DB ⊗WBB+

cris,B/(β(σ) ·B + Fil2) //

DF ⊗WFB+

cris,F/(β(σ) · F + Fil2)

0 0

where both columns are exact and all of the horizontal maps are injective, with Lemma4.8.2(3) being used to show that the lowest horizontal map is injective. Now we know thatN(d)⊗β(σ) lies in the image of the middle horizontal map. Since all of the horizontal mapsare injective, N(d) ∈ DB. Therefore we see that N induces a map N : DB → DB as desired.This endomorphism will satisfy the conditions of the proposition because it does so afterextending scalars to F = B[1/πB].

169

4.10. Moduli Space of Semistable Representations

Because this section deals entirely with coefficient rings being finite Qp-algebras, there isno fundamentally new content. We simply reprise [Kis08, §2.5].

Our goal here is to show that Ast is the maximal quotient of A over which the represen-tation VA is semistable with Hodge-Tate weights in [0, h], in the sense that for any A-algebraB which is finite as a Qp-algebra, the representation VA ⊗A B is semi-stable if and only ifA→ B factors through Ast. In order to prove this, we recall the following relations betweenweakly admissible filtered (ϕ,N)-modules and S-lattices of finite E-height, due to Kisin[Kis06].

Theorem 4.10.1 ([Kis06], e.g. Corollary 1.3.15). Let D be a weakly admissible filtered(ϕ,N)-module with Fil0D = D and Filh+1 D = 0. Then there is a finite free S[1/p]-moduleM and a map ϕ∗(M)→M whose cokernel is killed by E(u)h such that

(1) There is a canonical ϕ-equivariant isomorphism M/uM∼→ D.

(2) If M := M⊗S[1/p] O, then M admits a unique logarithmic connection

∇ :M→M⊗O Ω1O[1/uλ]

such that ∇ ϕ = ϕ ∇ and induces a differential operator

N∇ :M→ cM, m 7→ 〈∇(m),−uλ ddu〉

such that N∇|u=0 = N .(3) M admits a lattice, i.e. there exists a finite free S-module M which spans M and

such that the cokernel of 1⊗ ϕ : ϕ∗(M)→M is killed by E(u)h.

There is also a Γ∞-equivariant isomorphism

(4.10.2) HomS[1/p],ϕ(M,Sur[1/p])∼−→ HomFil,ϕ,N(D,B+

st)

which is constructed using some maps that have already appeared in this text (see [Kis08,p. 18] for the recipe). There is also an isomorphism

(4.10.3) HomB+cris,Fil,ϕ(D ⊗K0 B

+cris, B

+cris)

∼−→ HomFil,ϕ(D,B+cris)

∼−→ HomFil,ϕ,N(D,B+st)

and an isomorphism

(4.10.4) HomS[1/p],ϕ(M,Sur[1/p])∼−→ HomFil,ϕ(D,B+

cris)

constructed as in [Kis08, p. 18].We now show that the candidate Ast of Proposition 4.9.11 for the moduli space of semi-

simple representations satisfies this property.

Proposition 4.10.5 ([Kis08, Proposition 2.5.4]). Assume that A = (A)≤h. Let B bea finite Qp-algebra, ζ : A → B a map of Qp-algebras, and VB := VA ⊗A B. Then ζ factors

through Ast if and only if VB is semistable as a representation of Γ over Qp.

Proof. Suppose that ζ factors through Ast. Then Proposition 4.9.11 implies that (4.9.7)

is a Γ-equivariant map

DB ⊗Qp B+st∼= DB ⊗WB

B+st,B −→ V ∗B ⊗B+

st,B∼= VB ⊗Qp B

+st .

which is injective according to Lemma 4.9.9. We get an injection of Galois invariants

DB → (VB ⊗Qp B+st)

Γ

170

so that the dimension of the right side as a K0-vector space is at least as much as that of the

left side. But dimK0 DB = dimQp VB. Therefore dimQp VB ≥ dimK0(VB⊗Qp B+st)

Γ, so that VBis semistable.

Now suppose that VB is semistable. Let

DB := HomB[Γ](VB, B+st ⊗Qp B)

be the weakly admissible filtered (ϕ,N)-module associated to V ∗B. Denote by MB the S[1/p]-

module attached to DB according to the discussion at the beginning of §4.10. Let MB :=MA ⊗A B as usual, where MA was produced in Proposition 4.5.9 and MA := MA ⊗A A.Composing the map ι−1⊗AB of (4.9.1) with (4.10.2) and taking B-linear maps, we find that

(4.10.6) HomSB ,ϕ(MB,SurB )

∼→ VB∼→ HomB,Fil,ϕ,N(DB, B

+st,B)

∼→ HomSB ,ϕ(MB,SurB ).

Because S-lattices of height ≤ h are unique in MB = (Eur ⊗Qp V∗B)Γ by [Kis06, Propo-

sition 2.1.12], we may identify MB and MB. Using this identification and the maps in(4.10.6), Theorem 4.10.1 allows us to identify DB with DB = MB/uMB (WB-linearly andϕ-equivariantly), and then endow DB with an operator N coming from the operator on DB.

We then have a commutative diagram, where the top right horizontal arrow comes from(4.10.4).(4.10.7)

VB∼ //

id

HomSB ,ϕ(MB,SurB [1/p])

∼

∼ // HomB+cris,B ,Fil,ϕ(DB ⊗WB

B+cris,B, B

+cris,B)

VB

∼ // HomSB ,ϕ(MB,SurB [1/p]) // HomB+

cris,B ,Fil,ϕ(DB ⊗WBB+

cris,B, B+cris,B)

The maps in the top horizontal row are Γ-equivariant. Observing the diagram, it followsthat the same holds for the maps in the bottom row. The composite of the bottom row mapsinduces a Γ-equivariant map

(4.10.8) DB ⊗WBB+

cris,B → HomB(VB, B+cris,B).

We claim that this map is identical to that of (4.9.4). Since (4.9.4) is Γ-equivariant, it followsby Proposition 4.9.11 that ζ factors through Ast.

Theorem 4.10.9. As is standard in this section, let A be an algebra formally finitelygenerated over R, with a continuous action of Γ on a projective rank d A module VA. If his a non-negative integer, then there exists a quotient Ast,h of A such that

(1) If B is a finite Qp-algebra, and ζ : A → B a map of Qp-algebras, then ζ factorsthrough Ast,h if an only if VB = VA ⊗A B is semistable with Hodge-Tate weights in[0, h].

(2) There is a projective WAst,h-module DAst,h of rank d equipped with a Frobenius semi-linear automorphism ϕ and with a WAst,h-linear automorphism N such that for allζ : A→ B factoring through Ast,h, there is a canonical isomorphism

DB = DAst,h ⊗Ast,h B∼−→ HomB[Γ](VB, B

+st ⊗Qp B)

respecting the action of ϕ and N .

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Proof. Assume that VB is semistable with Hodge-Tate weights in [0, h]. Then VB isof E-height ≤ h: for we call VB of E-height ≤ h when the cokernel associated map of SB-modules ϕ∗(MB)→MB of Proposition 4.5.9(1) is killed by E(u)h. The proof of Proposition

4.10.5 identifies this S-module with another one, denoted MB, created from the (ϕ,N)module associated to VB. Now [Kis06, Lemma 1.2.2] associates the Hodge-Tate weights ofthe (ϕ,N)-module with the cokernel in the way that we require.

Because VB is of E-height ≤ h, we know that A → B factors through (A)≤h[1/p].Therefore we may replace A by the quotient (A)≤h defined in Proposition 4.5.5. Let Ast,h

be the ring Ast of Proposition 4.9.11 and set DAst := MA/uMA ⊗A Ast. If VB is semistablethen ζ factors through Ast by Proposition 4.10.5.

Conversely, if ζ factors through Ast then Proposition 4.10.5 implies that VB is semistable

of E-height ≤ h. If DB := (V ∗B⊗QpB+st)

Γ and MB is the S[1/p]-module associated to DB as inthe proof of Proposition 4.10.5, then the uniqueness of S-lattices of finite E-height [Kis06,

Proposition 2.1.12] implies that MB has E-height ≤ h, and the claim about Hodge-Tateweights once more follows from [Kis06, Lemma 1.2.2].

To see (2), concatenate the Γ-equivariant isomorphisms

(4.10.10) VB∼−→ HomB+

cris,B ,Fil,ϕ(DB ⊗WBB+

cris,B, B+cris,B)

∼−→ HomB,Fil,ϕ,N(DB, B+st,B),

where the first isomorphism appears on the top line of (4.10.7) and is deduced from in(4.10.3) by taking B-linear maps. Now (2) follows from applying HomB[Γ](−, B

+st,B) and the

fact that this functor is inverse to HomB,Fil,N,ϕ(−, B+st,B).

4.11. Hodge Type

In this section we follow [Kis08, §2.6] and the erratum [Kis09b, §A.4] to construct aquotient of Ast,h corresponding to semistable representations with a specified p-adic Hodgetype. First we recall the notion of p-adic Hodge type. For this, we fix an finite extensionfield E of Qp and suppose that A admits the structure of an E-algebra.

Definition 4.11.1. Suppose we are given a finite dimensional E-vector space DE witha filtration of DE,K := DE ⊗Qp K by E ⊗Qp K-submodules such that the associated graded

is concentrated in degrees [0, h]. Let v := DE,FiliDE,K , i = 0, . . . , h.

If B is a finite E-algebra and VB ∈ RepdΓ(B) such that VB is a de Rham representation,

then we say that VB is of p-adic Hodge type v if the Hodge filtration on the associatedfiltered (ϕ,N)-module has the same graded degrees as v. That is, VB has all its Hodge-Tateweights in [0, h] and for i = 0, . . . , h there is an isomorphism of B ⊗Qp K-modules

gri HomB[Γ](VB, BdR ⊗Qp B)∼−→ griDE,K ⊗E B.

Theorem 4.11.2 ([Kis08, Corollary 2.6.2]). With v as above, there exists a quotientAst,v of Ast corresponding to a union of connected components of SpecAst with the followingproperty. If B is a finite E-algebra and ζ : A → B is a map of E-algebras, then ζ factorsthrough Ast,v if and only if VB = VA ⊗A B is semistable of p-adic Hodge type v.

Proof. To begin with, we establish some notation. Let

(4.11.3) Fili ϕ∗(MA) = (1⊗ ϕ)−1(E(u)iMA) ⊂ ϕ∗(MA).

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The second part of Lemma 4.7.1 identifies DB ⊗WBOB and ϕ∗(MB ⊗SB OB in a formal

neighborhood of the ideal (E(u)) ⊂ OB. In particular, we have a ϕ-compatible identification

DB ⊗K0 K∼−→ ϕ∗(MB)/E(u)ϕ∗(MB).

From here we equip DB ⊗K0 K with a filtration, setting

(4.11.4) Fili(DB ⊗K0 K) := Fili ϕ∗(MA)/(E(u)ϕ∗(MA) ∩ Fili ϕ∗(MA))⊗A B.Next we check that this filtration on DB ⊗K0 K coincides with the one induced by the

isomorphism of Theorem 4.10.9(2), namely

DB∼−→ HomB[Γ](VB, B

+st ⊗Qp B.

Write DB for DB equipped with the filtration from Theorem 4.10.9(2). This is the standard

weakly admissible (ϕ,N) module over B attached to V ∗B. Now let MB be the S[1/p]-module

attached to DB as summarized in Theorem 4.10.1. The uniqueness of lattices of E-height≤ h implies that MB may be identified with MA ⊗A B. We conclude the proof by recalling[Kis06, 1.2.6-1.2.7], which reconstructs the filtration on DB⊗K0 K from MB as the preimage

filtration on DB ⊗WBOB the filtration defined in (4.11.3) (with MB in the place of MA)

under the map ξ of Lemma 4.7.1, specialized at OB/E(u)OB. This is precisely the same asthe filtration of (4.11.4), as desired.

Notice that we have identified Fili ϕ∗(MA)/(E(u)ϕ∗(MA)∩Fili ϕ∗(MA))⊗AB, which wasoriginally used to define the filtration on DB ⊗K0 K, with the a priori projective WB ⊗K0 K-module Fili(DB⊗K0 K). Since this is valid for all finite local E-algebras B, this implies thatFili ϕ∗(MA)/(E(u)ϕ∗(MA)∩Fili ϕ∗(MA)) is a projective A-module. Moreover, the discussionabove shows that, over A-algebras B that are finite E-algebras, the graded componentsof these modules determine the Hodge type of the associated Galois representation VB =VA⊗AB. Since projective modules have locally constant rank, this shows that the points ofSpecA corresponding to a given hodge type v form a union of connected components, whosecoordinate ring we will denote by Av. Namely, these are points p of SpecA such that fori = 0, 1, . . . , h, there is an isomorphism of WAp ⊗K0 K-modules

Fili ϕ∗(MA)/(E(u)ϕ∗(MA) ∩ Fili ϕ∗(MA))⊗A Ap∼−→ FiliDE,K ⊗E Ap.

Let Ast,v := Ast,h ⊗A Av.

4.12. Galois Type

In this section we further stipulate that B is local with residue field E, so that it is afinite, local E-algebra with residue field E. Let VB ∈ Repd

Γ(B). Following [Fon94], set

D∗pst(VB) = lim−→K′

HomB[ΓK′ ](VB, Bst ⊗Qp B),

where K ′ runs over finite field extensions of K.Let K0 ⊂ K denote the maximal unramified extension of K0, and let Γ0 ⊂ Γ be the inertia

group of Γ. Then D∗pst(VB) is a B ⊗Qp K0-module with a Frobenius semi-linear Frobenius

automorphism ϕ, a nilpotent endomorphsm N such that pϕN = Nϕ, and a B⊗Qp K0-linear

action of Γ0 which has open kernel and commutes with ϕ and N .We claim that D∗pst(VB) is finite and free as a B⊗Qp K0-module. We will show this using

the fact that ϕ is an automorphism and following the line of reasoning of [Kis09c, Lemma1.2.2(4)]. Firstly, we know that this module is finite and flat over B ⊗Qp K0 by definition

173

of the functor. To check that D∗pst(VB) is free, we need only check that the fibers over the

residue fields, i.e. the points of SpecE ⊗Qp K0, are of constant rank. This module arises by⊗K′0K0 from a free, rank d (ϕ,N)-module D over E⊗QpK

′0, where K ′/K is a finite extension

making VB semistable as a representation of ΓK′ . Since an unramified base change cannotmake “potentially semistable” into “semistable,” we may assume that K ′0 = K0. For anyunramified extension L0/K0, now let K ′0 (resp. L′0) denote K0 ∩ E (resp. L0 ∩ E), and alsolet E0 be the maximal subfield of E unramified over Qp. We observe that ϕ permutes thefactors labeled by µ of the decomposition

E ⊗Qp L0∼=∏µ

E ⊗Qp K0,

where µ runs over the set of embeddings µ : E0 → L′0 fixing K ′0. This shows that ϕ willpermute the factors of D⊗K0 L0 under this decomposition by µ, and each of these factorsis free of rank d. Therefore D∗pst(VB) is free of rank d as a B ⊗Qp K0-module, as desired.

Since the action of Γ0 commutes with the action ϕ, the traces of elements of Γ0 arecontained in B, and D∗pst descends to a representation of Γ0 on a finite free B-module PB.Because characteristic zero representations of finite groups are rigid, i.e. the deformationsof an action of a finite group on a vector space over a characteristic zero field E to artinianE-algebras arise by extension of scalars, this representation must be an extension of scalarsfrom a representation PB of Γ0 over E.

We have associated to a potentially semistable d-dimensional representation VB of Γ overB a representation of the inertia group of K over E which reflects the failure of VB to besemistable. We will call this the “Galois type” of VB, as follows.

Fix an algebraic closure Qp of Qp.

Definition 4.12.1. Let T : Γ0 → GLd(Qp) be a representation with open kernel. Wesay that VB is potetially semistable of type T provided that PB defined above is isomorphicto T .

Because we are working over a characteristic 0 field, it is equivalent to say that for anyγ ∈ Γ0, the trace of T (γ) is equal to the trace of γ on D∗pst(VB).

Our goal is to find a moduli space for Galois representations that are both potentiallysemistable and have Galois type T . Before we give a supporting lemma, we recall that theelement t ∈ Acris ⊂ B+

dR, which generates the maximal ideal of B+dR, is used in the definitions

Bst = B+st [1/t] and Bst,A = B+

st,A[1/t] (see §4.6).

Lemma 4.12.2 (Following [Kis08, Lemma 2.7.1]). For i ≥ 0 there is an isomorphism

WA · ti∼→ HomA[Γ](A(i), B+

st,A)

induced by multiplication by p−ri for ri a positive integer defined below, where A(i) denotes

A with Γ acting via the ith power of the p-adic cyclotomic character χ. In particular, if

Bst,A := B+st,A[1/t], then BΓ

st,A = WA.

This proof was done for local A in [Kis08], based on the well known case when A is afinite flat Zp-algebra. The general case requires additional notions, much along the lines ofLemma 4.8.2(5).

Proof. First we will show, following [Kis08], that any element x ∈ B+st,A such that Γ

acts on x via χi lies in B+cris,A. We may represent x as x =

∑ni=0 ai`

iu where ai ∈ B+

cris,A. As

174

the lemma is well known for A finite over Zp, one can apply Lemma 4.8.2(5) to concludethat ai = 0 for i > 0. Therefore, replacing x with a multiple of itself by a power of p, we seethat it suffices to prove that x ∈ Acris,A such that Γ acts on it via χi lies in

(4.12.3) WA ·ti

pri⊂ Acris,A ,

where ri is the largest non-negative integer such that ti/pri ∈ B+cris lies in Acris.

In the case that A is a complete local Noetherian Zp-algebra with finite residue field, thisLemma was proved in [Kis08]. We will reduce the proof to this case, and then recapitulatethe proof from [Kis08] afterwards.

First we note that if ri is the integer defined for (4.12.3), then the cokernel of the mapof Zp-modules

(4.12.4)

W −→ Acris

x 7→ x · ti

pri

is torsion-free, and therefore also flat as a Zp-module. Since ri is chosen to saturate thesub-Zp-module W · ti ⊂ Acris, this is clear enough: choose a representative y ∈ Acris of anonzero element of the cokernel of this map. If y is a torsion element of the cokernel, thenthere exists some positive integer n such that pn · y ∼= x · ti

prifor some x ∈ W but pn−1 · y

does not lie in W · tipri

. But since Acris is a flat Zp-module, this would imply that x is a unit

in W , and without loss of generality x = 1. But then pn−1y ∼= ti

pri+1 ∈ Acris, a contradiction.

Secondly, we note that the image of (4.12.3) lies in the submodule Acris⊗ZpA of Acris,A .

Recall that since Acris is p-adically complete, the natural map Acris⊗ZpA → Acris,A is indeed

an inclusion. Also observe that the natural map W ⊗Zp A → WA is an isomorphism: since

W/Zp is finite, W ⊗Zp A is mRA-adically complete. Therefore we can factor the inclusion

(4.12.3) as the composition of natural inclusions

(4.12.5)

W ⊗Zp A → Acris ⊗Zp A

x⊗ y 7→ x · ti

pri⊗ y

followed by the inclusionAcris ⊗Zp A

→ Acris,A .

Recall from the proof of Lemma 4.8.2(5) the following notions: let M be the set ofmaximal ideals of Spf(A) as a Spf(Zp)-formal scheme, corresponding to maximal ideals ofA/mRA

. In the proof, we showed that the natural maps

(4.12.6) A →∏m∈M

Am, Acris,A →∏m∈M

Acris,Am

are injective. Of course, everything in the discussion about (4.12.5) applies to Am in theplace of A, so that for each m ∈M there are maps

(4.12.7) W ⊗Zp Am → Acris ⊗Zp A

m

factoring WAm∼= W ⊗Zp A

m → Acris,Am . Therefore, assuming the result of this lemma when

A is a complete local ring with finite residue field, we can deduce the general case (i.e.

175

A not necessarily local) from the truth of the result over complete local rings Am of A asfollows.

Consider the inclusions

W ⊗Zp A //

(4.12.5)

∏m∈M W ⊗Zp A

m

(4.12.7)

Acris ⊗Zp A

(4.12.6)//∏

m∈M Acris ⊗Zp Am

We will be done if we can show that the image of W ⊗Zp A in the bottom right is the

intersection of the images of (4.12.7) and (4.12.6). Using the fact that the cokernel of(4.12.4) (and therefore the cokernel of the vertical maps as well) is flat, we apply Lemma4.12.8 to draw this conclusion and finish the proof.

We have finished the deduction of the proof of the lemma in the general case from theproof in the case that A is local. It remains to prove the case that A is local, reprising[Kis08].

Let A be a complete local Noetherian Zp-algebra with finite residue field F and maximal

ideal m. Let x ∈ B+st,A such that Γ acts on it by χi. The beginning of this proof has reduced

our remaining work to the case that x ∈ Acris,A ; we must show that x is in the image of(4.12.3).

Let q1 ⊃ q2 ⊃ · · · be a decreasing sequence of ideals of A such that ∩∞j=1qj = 0 andA/qj is a finite W (F)[1/p]-algebra. Let qj := A ∩ qj. Then for each m ≥ 0, it follows that

we have qj ⊂ mmA for large enough j. Since A is m-adically complete, A

∼→ lim←−j A/qj .

Moreover, (qj) is a sequence of ideals of definition for the topology on A. Therefore

Acris,A∼→ lim←−

j

Acris,A/qj.

The same is true with W in place of Acris.Using the integer ri defined for (4.12.3), then for all j ≥ 1, the image of x in Acris,A/qj

is contained in the image of WA · ti

priin Acris,A/qj

because the lemma is known for A finite

over Zp. This property is stable under taking the inverse limit indexed by j, so that we

conclude that x ∈ WA · ti

prias desired.

Lemma 4.12.8. Let R be a commutative ring and let M,N, S, and T be flat R-modules.Fix injective maps M → N , S → T such that M is a pure submodule of N , i.e. the cokernelof the inclusion is flat. Consider N ⊗R S, M ⊗R T , and M ⊗R S as submodules of N ⊗R Tunder the natural inclusion maps. Then

N ⊗R S ∩M ⊗R T = M ⊗R S.

176

Proof. Let L be the cokernel of M → N , which is a flat R-module. Let U be thecokernel of S → T . Then we have the following diagram of exact sequences:

0

0

0 // M ⊗R S

// M ⊗R T

// M ⊗R U

// 0

0 // N ⊗R S

// N ⊗R T

// N ⊗R U

// 0

0 // L⊗R S

// L⊗R T

// L⊗R U

// 0

0 0 0

Let x be an element of N ⊗R S which is in the intersection described in the statement ofthe lemma. Then the image of x in L⊗R T is 0, so that the image of x in L⊗R S is also 0.Therefore x is in the image of M ⊗R S in N ⊗R S, as desired.

Now we show that the theory of (ϕ,N)-modules with coefficients functions as expected.

Proposition 4.12.9 (Following [Kis08, Proposition 2.7.2]). Suppose that A = Ast,h.Then the map

(4.12.10) DA ⊗WABst,A −→ HomA(VA, Bst,A)

induced from (4.9.7) by setting B = A and tensoring by ⊗B+st,ABst,A is an isomorphism. In

particular,

(4.12.11) DA∼−→ HomA[Γ](VA, Bst,A).

This proof requires small modifications from that of [Kis08].

Proof. Lemma 4.9.9 tells us that (4.12.10) is an injection. Furthermore, because A =Ast,h, Theorem 4.10.9 tells us that the right hand side and left hand side of (4.12.10) arefinite free Bst,A-modules of the same rank. Therefore it will suffice to show that this mapinduces an isomorphism on top exterior powers, and we freely restrict ourselves to the casethat VA is free of rank 1 over A. We note that in either of these cases, VA arises byextension of scalars from a complete local Noetherian ring. This is the case because theuniversal moduli space of 1-dimension representations of Γ is semi-local, and in particular,the underlying scheme is the disjoint union of spectra of finite fields. Therefore, once weshow that (4.12.10) is stable under extension of coefficients in a sense we will define in amoment, we can resort with no concern to the arguments of [Kis08], which are working inthe case that A is a complete Noetherian local ring with finite residue field.

We will now show that the property that (4.12.10) is an isomorphism is preserved byextension of coefficients which are adic R-algebras, in a sense we now define. This extendsan observation made at the beginning of the proof of [Kis08, Proposition 2.7.2]. Let A → A′

be a map in the category of adic R-algebras, and write A′ := A′[1/p] as usual. We claimthat the map (4.12.10) for VA′ , i.e. the representation arising by extension of scalars fromVA′ := VA ⊗A A′, is obtained from (4.12.10) by extending scalars by ⊗Bst,A

Bst,A′ . To see

177

this, we observe that each of the factors of the map (namely (4.9.6), (4.9.2), which arises fromProposition 4.5.9, and the map ξ of Lemma 4.7.1) are compatible with the scalar extensionprocess.

1-dimensional semistable representations are crystalline and crystalline characters are theproduct of an unramified character and a Lubin-Tate character2 determined by the Hodgefiltration. Therefore, VA|Γ0

is locally constant on SpecA because, according to Theorem4.11.2, the Hodge type is constant on connected components of SpecA. Replacing SpecAwith one of its connected components, we may assume that VA|Γ0

∼ θ|Γ0, where θ is the

product of conjugates of Lubin-Tate characters. It will suffice to prove the proposition intwo cases, VA ∼ θ and VA an unramified character. This is the case because we may tensorthe factors in (4.12.10) for VA ∼ θ with the factors for VA unramified to derive the generalcase.

If VA ∼ θ, then VA arises by extension of scalars from a representation valued in thering of integers of a finite extension of Qp. The observation about extension of scalars givenabove now allows us to assume that A is such a ring of integers. Therefore this case followsdirectly from Theorem 4.10.9(2), as A is finite as a Qp-algebra.

Now for the unramified case, the filtration on DA is trivial so h = 0. Let k be the residuefield of K. As a result (cf. [FO, §7.2.2]), the slope of DA is zero so that (4.9.4) arises byextension of scalars ⊗AB+

cris,A from an isomorphism

DA∼−→ HomA[Γ](VA,W (k)A),

and therefore is an isomorphism as well.Now we come to the second statement of the proposition. Lemma 4.12.2 gives us that

BΓst,A = WA, so that the usual regular G-ring formalism (e.g. [FO, §2]) will apply, and allow

us to conclude thatDA

∼−→ HomA[Γ](VA, Bst,A).

As elements of the image of DA under (4.12.10) have image in B+st,A ⊂ Bst,A, (4.12.11)

follows.

Recall that we have fixed E as a finite extension of Qp such that A admits the structureof a E-algebra. Let v be a p-adic Hodge type as in Definition 4.11.1. We fix a representation

T : Γ0 → EndE(DE)∼−→ GLd(E).

Theorem 4.12.12 ([Kis08, Theorem 2.7.6]). There exists a quotient AT,v of A such thatfor any finite E-algebra B, a map of E-algebras ζ : A→ B factors through AT,v if and onlyif VB = VA ⊗A B is potentially semistable of Galois type T and p-adic Hodge type v.

Proof. Let L/K be a finite Galois extension such that the inertia subgroup IL ⊂ IKis contained in kerT . The group change map along with Theorem 4.11.2 give us a quotientApst,v of A such that ζ factors through Apst,v if and only if VB|ΓL is semistable of p-adic

Hodge type v. Assume A = Apst,v from now on.Let WL denote the ring of integers of L0, the maximal unramified subfield of L. Set

WL,A := (WL)A. Proposition 4.12.9 gives us an isomorphism of finite free WL,A-modules

DA∼−→ HomA[ΓL](VA, B

+st,A)

2See e.g. [Ser68, III.A.4] for a discussion of Lubin-Tate characters.

178

that is compatible with the natural action of ϕ. The Galois group Gal(L/K) acts L0-semi-linearly on HomA[ΓL](VA, B

+st,A), and the inertia group IL/K ⊂ Gal(L/K) acts L0-linearly

(cf. [FO, Prop. 6.58]). Since the action of Gal(L/K) commutes with ϕ, if σ ∈ IL/K , then thetrace Tr(σ) is in (WL,A)ϕ=1 = A. Because characteristic 0 representations of finite groupsare rigid, Tr(σ) is a locally constant function on SpecA. Denote by AT,v the quotient ofA corresponding to the union of components of SpecA where Tr(σ) = Tr(T (σ)) for all

σ ∈ Γ0.

Corollary 4.12.13 ([Kis08, Corollary 2.7.7]). There exists a quotient AT,vcr of A suchthat for any finite E-algebra B, a map of E-algebras ζ : A→ B factors through AT,vcr if andonly if VB = VA ⊗A B is potentially crystalline of Galois type T and p-adic Hodge type v.

Proof. Theorem 4.10.9 provides for us a finite projective WAT,v-module DAT,v equippedwith a linear endomorphism N . We know that VB is potentially crystalline if and only if thespecialization of N by ζ to B vanishes. Therefore we may take AT,vcr to be the quotient ofAT,v defined by the relation N = 0.

4.13. Final Remarks

Combining the results of Chapter 3 (see Theorem 3.2.5.1) with Chapter 4, we have

several the following results. Let Γ is the absolute Galois group of a finite field extensionK of Qp, and choose a residual pseudorepresentation D. Recall that PsRD = Spf BD is the

deformation space of D, which is Noetherian since Γ is finitely generated. Also recall thatRepD denotes the groupoid of Azumaya algebra-valued continuous representations of Γ withconstant residual pseudorepresentation D. The natural map

ψ : RepD → PsRD

is universally closed, formally of finite type, and is the algebraization of a finite type SpecBD-algebraic stack RepE(R,Du

D),D, where E(R,Du

D) is the universal Cayley-Hamilton BD-algebra

E(R,DuD) := (Zp[[Γ]]⊗Zp BD)/CH(Du

D),

which is finitely generated as a BD-module.Here are a few observations regarding the implications of what we have proved.

Observation 4.13.1. Combining the algebraicity of RepΓ over PsRΓ with the projec-tivity of the moduli of Kisin modules L≤h over RepΓ, the moduli of Kisin modules is alge-braizable over PsRΓ and universally closed, with projective PsRΓ-subschemes.

Observation 4.13.2. Let A denote the admissible coefficient Zp-algebra of a continuousA-Azumaya algebra-valued representation

ρ : Zp[[Γ]]⊗Zp A −→ E

of Γ with constant residual pseudorepresentation D, which we can assume to be formallyfinitely generated over BD (i.e. the quotient of a restricted power series over BD in finitelymany indeterminates). For example, one can think of A as the universal coefficient sheafof rings ORepD

. Chapter 4 constructs closed subspaces Xcond = Spec(A[1/p])/Icond of the

Noetherian Jacobson schemeX := SpecA[1/p], which are precisely the loci of specializationsof ρ to A-algebras, finite as Qp-algebras, satisfying certain conditions from p-adic Hodgetheory. Now consider the algebraization statement over PsRD. It implies that there exists

179

a universal finite type SpecBD-algebraic stack RepE(R,DuD

),D such that ρ arises from the

universal representation of E(R,DuD

) over RepE(R,DuD

),D by pullback along some morphism

fρ : SpecA −→ RepE(R,DuD

),D.

(We have set R := Zp[[R]].) Therefore there is a finitely generated BD-subalgebra Afg of A

with a (BD-typically) continuous representation

ρfg : E(R,DuD)⊗BD A

fg → Efg

such that ρ ' ρfg ⊗Afg A. We can now consider the closed subscheme of SpecAfg[1/p]corresponding to the condition “cond” from p-adic Hodge theory: it is cut out by the idealthat is the quotient of the composite map

Afg[1/p] −→ A[1/p] −→ A[1/p]/Icond.

This is an example of finite type SpecBD[1/p]-schemes which are universal moduli spacesfor representations of the module finite BD[1/p]-algebra E(R,Du

D)[1/p] which are required

to satisfy a p-adic Hodge theoretic condition.

Observation 4.13.3. One can make sense of the notion of a K-valued pseudorepresen-tation D of Γ satisfying or not satisfying certain conditions from p-adic Hodge theory, whereK is a finite field extension of Qp. After a finite extension of K, D is realizable as thedeterminant of a semisimple K-valued representation ρssD . Then one can say that D has ap-adic Hodge theoretic property if ρssD does. Of course, this does not imply that all represen-

tations of Γ with semisimplification ρssD (i.e. representations in the fiber of ψ over D) havethis property.

Observation 4.13.4. Since φ is universally closed, the constructions above give a closedsubspace of PsRD[1/p] corresponding to certain p-adic Hodge theory conditions, say cut outby an ideal Icond

PsR ⊂ BD[1/p]. As a result, one can construct a quotient

E(R,DuD)cond := E(R,Du

D)[1/p]⊗BD[1/p] BD[1/p]/IcondPsR

through which all representations satisfying this condition must factor. Conversely, it seemsthat its semisimple, p-adic field-valued representations must satisfying the condition, as longas they induce pseudorepresentations parameterized by SpecBD[1/p]. This construction mayeven be able to be refined if representations satisfying this condition are shown to cut outappropriately linear subspaces of the projective spaces of extensions described in §2.2. Inthis case, there should exist a quotient algebra of E(R,Du

D)cond whose representations (given

that they are parameterized by SpecBD[1/p]) are precisely those satisfying the condition.

We are curious if there is any useful application of these observations.

180

APPENDIX A

A Remark on Projective Morphisms

There are several notions of projectivity of a morphism of schemes. We will use thefollowing terminology.

Definition A.1 ([Sta, Definition 01W8]). Let f : X → S be a morphism of schemes.

(1) We say f is projective if X is isomorphic as an S-scheme to a closed subscheme ofa projective bundle P(E) for some quasi-coherent finite type OS-module E .

(2) We say that f is H-projective if there exists an integer n and a closed immersionX → PnS over S.

(3) We say that f is locally projective if there exists an open cover of S such that therestriction of f to each element of the cover is projective.

Example A.2. A finite morphism is always projective, but is not always H-projective.

Local projectiveness and local H-projectiveness are equivalent. Though H-projectivity ispreserved under composition using the Segre embedding ([Sta, Lemma 01WE]), this propertyof projectivity requires quasi-compactness of the base [Vak12, Exercise 18.3.B]. Projectivityis not a local property on the base. However, given a (very) ample line bundle for a projectivemorphism, one can check projectivity locally when the base is locally Noetherian [Vak12,Exercise 18.3.G].

We will use “projective” morphisms so that we can prove projectivity of a morphismlocally on the base (as long as we can glue together the ample line bundle).

181

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